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Product formulas and convolutions for two-dimensional

Laplace-Beltrami operators: beyond the trivial case

Rúben Sousa ∗ Manuel Guerra † Semyon Yakubovich ‡

June 26, 2020

Abstract

We introduce the notion of a family of convolution operators associated with a given elliptic partial

differential operator. Such a convolution structure is shown to exist for a general class of Laplace-Beltrami

operators on two-dimensional manifolds endowed with cone-like metrics. This structure gives rise to a

convolution semigroup representation for the Markovian semigroup generated by the Laplace-Beltrami

operator.

In the particular case of the operator L = ∂2x +

1

2x∂x +

1

x∂2θ on R

+× T, we deduce the existence of a

convolution structure for a two-dimensional integral transform whose kernel and inversion formula can

be written in closed form in terms of confluent hypergeometric functions. The results of this paper can

be interpreted as a natural extension of the theory of one-dimensional generalized convolutions to the

framework of multiparameter eigenvalue problems.

Keywords: Laplace-Beltrami operator, generalized convolution, product formula, eigenfunction ex-

pansion, convolution semigroup, multiparameter eigenvalue problem.

1 Introduction

The problem of constructing generalized convolutions associated with a given Sturm-Liouville operator

L has been widely studied in the literature. Such generalized convolutions, whose defining property is

that they should trivialize the eigenfunction expansion of L in the same way as the ordinary convolution

trivializes the Fourier transform, allow one to develop the basic notions of harmonic analysis in parallel

with the standard theory. Among other applications, this construction yields a convolution semigroup

representation for the kernel of the heat semigroup {e−tL}t≥0; in other words, it enables us to interpret

the diffusion process generated by L as a generalized Lévy process. It is known that such convolution

structures exist for a wide class of Sturm-Liouville operators on bounded or unbounded intervals [5, 43, 44].

It is quite natural to wonder if one can also construct generalized convolutions associated with elliptic

operators in spaces of dimension d > 1. Indeed, the heat semigroup {e−t∆}t≥0 generated by the Laplacian

on Rd is a convolution semigroup with respect to the ordinary convolution on Rd, and this suggests that

it may be possible to devise a similar property for other elliptic operators on subsets of Rd or on d-

dimensional Riemannnian manifolds. However, this task turns out to be considerably more difficult than

in the one-dimensional setting, as we now explain.

Let M be a Riemannian manifold and m a positive measure on M . Consider a self-adjoint elliptic par-

tial differential operator L on L2(M,m) whose eigenfunction expansion (cf. [21], [16, Theorem XIV.6.6])

∗Corresponding author. CMUP, Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do

Campo Alegre 687, 4169-007 Porto, Portugal. Email: rubensousa@fc.up.pt†ISEG – School of Economics and Management, Universidade de Lisboa; REM – Research in Economics and Mathemat-

ics, CEMAPRE, Rua do Quelhas 6, 1200-781 Lisbon, Portugal. Email: mguerra@iseg.ulisboa.pt‡CMUP, Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687,

4169-007 Porto, Portugal. Email: syakubov@fc.up.pt

1

is of the form (Fh)k(λ) =∫M h(ξ)ωk,λ(ξ)m(dξ) (λ ∈ R, k = 0, 1, . . .). The basic requirement that a

generalized convolution ⋆ associated with L should satisfy is the following [44, 47, 49]: ⋆ is a bilinear

operator on the space of finite complex measures on M such that for any two such measures µ, ν we have

∫

M

ωk,λ d(µ ⋆ ν) =(∫

M

ωk,λ dµ)·(∫

M

ωk,λ dν)

(λ ∈ R, k = 0, 1, . . .). (1.1)

In particular, the (generalized) eigenfunctions ωk,λ should satisfy the product formula

ωk,λ(ξ1)ωk,λ(ξ2) =

∫

M

ωk,λ dπξ1,ξ2 (1.2)

where the measures πξ1,ξ2 = δξ1 ⋆ δξ2 do not depend on λ and k. Usually one also requires that these are

positive measures, so that the convolution is positivity-preserving. In certain specific cases where the ωk,λcan be written in terms of classical special functions, a closed-form expression for the product formula

measures πξ1,ξ2 has been determined [4, 5, 28, 29, 33, 42, 46]; however, the generality of such results

is severely limited. In the one-dimensional setting, a general theorem on the existence of such product

formulas has been established via a PDE technique [44, 51] which consists in studying the properties of

the hyperbolic equation Lξ1u = Lξ2u which is satisfied by the product of the eigenfunctions (here Lξdenotes the differential operator acting on the variable ξ). But this technique does not extend to the

multidimensional case, because the equation Lξ1u = Lξ2u becomes ultrahyperbolic and the corresponding

Cauchy problem becomes ill-posed (see e.g. [13]). The challenge is therefore to understand which extra

assumptions need to be imposed to make it possible to prove the existence of the product formula (1.2) for

the eigenfunctions of general elliptic partial differential operators. (Here the meaning of ‘general’ is that

one should include operators for which closed-form expressions for the eigenfunctions are not available.)

The goal of this paper is to construct product formulas and convolutions for elliptic operators on

two-dimensional manifolds M which admit separation of variables in the sense that the eigenfunctions

are of the form ωk,λ(ξ) = ψk,λ(x)φk,λ(y) (ξ = (x, y) ∈M).

To outline the ideas behind our construction, we first observe that if M = M1 ×M2 is a product

of Riemannian manifolds endowed with the product metric (so that the Laplace-Beltrami operator on

M1 ×M2 obviously admits separation of variables) and if there exists a convolution for the Laplace-

Beltrami operator on both M1 and M2, then we can trivially define a convolution associated with the

Laplace-Beltrami operator on M by taking the product of the convolutions on M1 and M2 (see Example

5.3). Here we focus on manifolds of the form M = R+0 ×M2 (R+

0 = [0,∞)) where instead of the product

metric structure we consider Riemannian structures defined by the so-called cone-like metrics, i.e. possibly

singular metrics of the form g = dx2+A(x)2gM2 ; note that this metric structure is a generalization of the

metric cone [11]. In the body of the paper we restrict ourselves to settings where M2 is one-dimensional,

but this is not an essential restriction. The Laplace-Beltrami operator of (M, g) is

∆ = ∂2x +A′(x)

A(x)∂x +

1

A(x)2∆2

where ∆2 is the Laplace-Beltrami operator of M2. The operator ∆ admits separation of variables: its

eigenfunctions ωk,λ can be written as the product of eigenfunctions of ∆2 and eigenfunctions of Sturm-

Liouville operators of the form ∆1,k := d2

dx2 + A′(x)A(x)

ddx − ηk

A(x)2 , where ηk ≥ 0 is a separation constant.

One of our contributions is to determine conditions on the function A which ensure that a product

formula exists for the eigenfunctions of each of the operators ∆1,k. If the eigenfunctions of ∆2 also admit

a product formula, then this gives rise to a product formula for ωk,λ which is analogous to (1.2), but

with the difference that in general the measure of the product formula also depends on the separation

parameter k.

In fact, this dependence on k shows that the problem of constructing a convolution for ∆ has a positive

answer provided that the operator ⋆ (for which (1.1) should hold) is allowed to depend on k. This naturally

leads to the notion of a family of convolutions associated with a given elliptic operator. These are in fact

families of hypergroups, because each of its convolutions endows the spaceM with a hypergroup structure,

2

as defined by Jewett [5, 24]. As we will see, many properties of the convolution algebras determined by

one-dimensional (generalized) convolutions can be extended to the families of convolutions discussed here.

In particular, such families provide a natural generalization of the convolution semigroup property for

the heat semigroup generated by the Laplace-Beltrami operator (see Corollary 4.6).

As noted above, our approach is applicable to a general class of elliptic operators whose eigenfunc-

tions are generally not expressible in terms of special functions. In addition, it allows us to recover as

particular cases the existence of product formulas and convolutions for certain (eigen)functions of hy-

pergeometric type. Our main example (Example 5.1) is the elliptic operator L = ∂2x + 12x∂x + 1

x∂2θ on

(0,∞) × T, which belongs to a family of Laplace-Beltrami operators on conic surfaces that has been

the object of recent studies [6, 7, 36]. We show that the eigenfunctions ωk,λ of L can be written

in terms of confluent hypergeometric (or Whittaker) functions, and that the eigenfunction expansion

(Fh)k(λ) =∫(0,∞)×T

h(x, θ)ωk,λ(x, θ)√x dxdθ admits a closed-form inversion formula. The proof of this

inversion formula relies on an apparently little known result on the spectral measure of the generator of

a drifted Bessel process. Using our general construction, we obtain the existence of positivity-preserving

convolutions ⋆k

associated with the two-dimensional integral transform F via the linearisation property

(F(h⋆kg))k = (Fh)k · (Fg)k; these convolutions give rise to a decomposition of the law of the diffusion

process generated by L in terms of the laws of drifted Bessel processes.

Our assumption that the elliptic operator L admits separation of variables is equivalent to requiring

that the eigenfunction equation −Lu = λu can be reduced to a multiparameter Sturm-Liouville eigenvalue

problem (see Remark 6.6). The main contribution of this paper can therefore be restated as follows: for a

general class of multiparameter Sturm-Liouville problems, it is possible to construct associated generalized

convolution structures in which the basic notions of harmonic analysis can be developed in analogy with

the classical theory. We hope that the results of this study are a first step towards a more general theory

of convolutions associated with multiparameter eigenvalue problems.

The structure of the paper is the following. Section 2 gives background on the eigenfunction expansion

of Laplace-Beltrami operators on M = R+0 × T endowed with cone-like metrics and on the spectral

representation of the heat kernel. The product formula for eigenfunctions of the Laplace-Beltrami operator

is established in Section 3, where we also define the associated family of convolutions and describe

the main continuity and mapping properties of the convolution structure. In Section 4 we introduce

the notion of infinitely divisible measures and convolution semigroups and we discuss the convolution

semigroup properties of operator semigroups determined by the Laplace-Beltrami operator. Some special

cases where the convolution is related with functions of hypergeometric type are presented in Section 5.

Finally, in Section 6 we show that the construction of the previous sections can be developed on manifolds

of the form M = R+0 × I, where I is an interval of the real line. The Appendix collects some known

results on one-dimensional generalized convolutions which are used throughout the paper.

Notation. In the sequel, R+ and R+0 stand for the positive and nonnegative numbers respectively. We

denote by C(E) the space of continuous complex-valued functions on a given space E; Cb(E), C0(E),

Cc(E) and Ck(E) denote, respectively, its subspaces of bounded continuous functions, of continuous

functions vanishing at infinity, of continuous functions with compact support and of k times continuously

differentiable functions. For −∞ ≤ a < b ≤ ∞, ACloc(a, b) is the space of locally absolutely continuous

functions on the interval (a, b). For a given measure µ on a measurable space E, Lp(E, µ) (1 ≤ p ≤ ∞)

is the Lebesgue space of complex-valued p-integrable functions with respect to µ. The indicator function

of a subset B ⊂ E is denoted by 1B(·). The set of all probability (respectively, finite positive, finite

complex) Borel measures on E is denoted by P(E) (respectively, M+(E), MC(E)), and we denote by

δx the Dirac measure at a point x. For given measures µ, µ1, µ2, . . . ∈ MC(E), we write µnw−→ µ

(respectively µnv−→ µ) if the measures µn converge weakly (resp. vaguely) to µ as n → ∞, i.e. if

limn→∞∫E g(ξ)µn(dξ) =

∫E g(ξ)µ(dξ) for all g ∈ Cb(E) (resp. for all g ∈ C0(E)).

3

2 The eigenfunction expansion of the Laplace-Beltrami operator

Throughout the paper we consider the (possibly singular) Riemannian manifold (M, g), whereM = R+0 ×T

(with T = R/Z) and the (C1, possibly non-smooth) Riemannian metric is given by

g = dx2 +A(x)2dθ2 (0 ≤ x <∞, θ ∈ T) (2.1)

where the function A is such that

A ∈ C(R+0 ) ∩ C1(R+), A(x) > 0 for x > 0,

∫ 1

0

dx

A(x)<∞,

A′

Ais nonnegative and decreasing.

(2.2)

The Riemannian volume form on M is dω =√det g dxdθ = A(x)dxdθ. Thus, the Riemannian gradient

of a function u :M −→ C is

∇u =

(∂xu,

1

A2∂θu

),

and the Laplace-Beltrami operator is

∆ = div ◦ ∇ = ∂2x +A′(x)

A(x)∂x +

1

A(x)2∂2θ . (2.3)

To introduce the closure of ∆ with reflecting boundary at x = 0, we proceed in the standard way (see

e.g. [39]). Consider the Sobolev space H1(M) ≡ H1(M,ω) ={u ∈ L2(M,ω) | ∇u ∈ L2(M,ω)

}, and the

sesquilinear form E : H1(M)×H1(M) −→ C, defined as

E(u, v) = 〈∇u,∇v〉L2(M,ω) =

∫

M

(∂xu∂xv +

1

A(x)2∂θu∂θv

)dω. (2.4)

It is clear from (2.4) that E is symmetric, positive semidefinite and, since its graph norm ‖u‖2Γ(E) =

‖u‖2L2(M,ω) + E(u, u) coincides with the norm of H1(M), it is a closed sesquilinear form. Let

DN ={u ∈ H1(M)

∣∣ ∃v ∈ L2(M,ω) such that 〈∇u,∇z〉 = −〈v, z〉 for all z ∈ H1(M)}.

The mapping u 7→ ∆Nu = v is a well defined linear operator in the domain DN . It follows from the

above that (∆N ,DN ) is a self-adjoint operator in L2(M,ω) ([39], Theorem 10.7), called the Neumann

Laplacian. It is an extension of the Laplace-Beltrami operator defined in a domain of smooth functions

satisfying the reflective boundary condition at x = 0

(A∂xu) (0, θ) = 0 ∀θ ∈ T.

We use the notations Lp(M) = Lp(M,ω), Lp(A) = Lp(R+, A(x)dx), and consider the Fourier decom-

position

L2(M) =⊕

k∈Z

Hk, Hk ={ei2kπθv(x)

∣∣ v ∈ L2(A)}, (2.5)

where Hk are regarded as Hilbert spaces with inner product⟨ei2kπθu, ei2kπθv

⟩Hk

= 〈u, v〉L2(A). The

direct sum is also regarded as a Hilbert space with inner product 〈{uk}, {vk}〉⊕Hk=

∑k∈Z

〈uk, vk〉Hk, such

that for u(x, θ) =∑k∈Z

ei2kπθuk(x), v(x, θ) =∑k∈Z

ei2kπθvk(x), we have

〈u, v〉L2(M) =∑

k∈Z

〈uk, vk〉L2(A)

E(u, v) =∑

k∈Z

Ek(uk, vk),

where Ek(uk, vk) =∫R+

(u′k(x)v

′k(x) +

(2kπ)2

A(x)2 uk(x)vk(x))A(x)dx are sesquilinear forms with domains

D(Ek) ={u ∈ L2(A) ∩ ACloc(R

+)∣∣∣ 2kπA

u ∈ L2(A), u′ ∈ L2(A)}.

4

Thus, we obtain the decomposition, compatible with (2.5):

H1(M) =⊕

k∈Z

D(Ek).

It can be checked that the forms Ek are symmetric, positive, and closed. Therefore, a similar argu-

ment allows us to construct self-adjoint realizations of the Sturm-Liouville operators ∆ku(x) = u′′(x) +A′(x)A(x) u

′(x) − (2kπ)2

A(x)2 u(x), whose domain is

D(∆k) ={u ∈ L2(A)

∣∣ u, u′ ∈ ACloc(R+), ∆ku ∈ L2(A), (Au′)(0) = 0

},

for k ∈ Z. This provides a decomposition of the Neumann Laplacian:

DN =⊕

k∈Z

D(∆k), ∆N

(∑

k∈Z

ei2kπθuk(x)

)=

∑

k∈Z

ei2kπθ∆kuk(x). (2.6)

The first step towards the construction of convolution operators associated with ∆ is the follow-

ing characterization of the separable solutions of the eigenfunction equation −∆u = λu with reflecting

boundary condition at x = 0.

Lemma 2.1. For each (k, λ) ∈ Z × C, there exists a unique solution wk,λ ∈ Hk,∞ :={ei2kπθv(x)

∣∣ v ∈C(R+

0 )}

of the boundary value problem

−∆u = λu, u(0, θ) = ei2kπθ , u[1](0, θ) = 0 (2.7)

where u[1](x, θ) = A(x) (∂xu)(x, θ). Moreover, λ 7→ wk,λ(x, θ) is, for each fixed (x, θ) ∈M and k ∈ Z, an

entire function of exponential type.

Proof. Clearly, any such solution must be of the form wk,λ(x, θ) = ei2kπθv(x), where v is a solution of

−∆kv(x) = λv(x), v(0) = 1, (Av′)(0) = 0. (2.8)

The result therefore follows from a standard existence and uniqueness theorem for solutions of Sturm-

Liouville boundary value problems (Lemma A.1).

The unique solution of (2.8) will be denoted by vk,λ(x), so that wk,λ(x, θ) = ei2kπθvk,λ(x). Throughout

the paper we will make frequent use of the normalized form of the function vk,λ which is described in the

following lemma (whose proof is elementary):

Lemma 2.2. Define vk,λ(x) :=vk,λ(x)ζk(x)

, where ζk(x) := cosh(2kπ

∫ x0

dyA(y)

). Then vk,λ(·) is a solution of

ℓk(v) = λv, v(0) = 1, (Bkv′)(0) = 0 (2.9)

where

ℓk(g) := − 1

Bk(Bkg

′)′, Bk(x) := A(x) ζk(x)2. (2.10)

Moreover, we haveB′

k

Bk= η + φ, where η(x) = 4kπ

A(x) tanh(2kπ∫ x0

dyA(y) ) ≥ 0 and the functions φ = A′

A and

ψ := 12η

′ − 14η

2 +B′

k

2Bkη = (2kπ)2

A2 are both decreasing and nonnegative.

The final assertion of the above lemma implies, in particular, that the assumption (A.6) of the

Appendix holds for the coefficients p = r = Bk of the Sturm-Liouville operator ℓk.

In what follows, to lighten the notation, points of M are denoted by ξ = (x, θ), ξ1 = (x1, θ1), etc.

It is well-known that the classical Weyl-Titchmarsh-Kodaira theory of eigenfunction expansions of

Sturm-Liouville operators can be generalized to elliptic partial differential operators on higher-dimensional

spaces, see e.g. [21], [16, Theorem XIV.6.6]. As remarked in [16, p. 1713], the knowledge about the

boundary conditions satisfied by the kernels of the eigenfunction expansion is much smaller in the (general)

5

multidimensional case, when compared to the one-dimensional setting. However, in the special case where

separation of variables can be applied to the eigenvalue problem for the elliptic operator and therefore

the eigenvalue equation reduces to a system of ordinary differential equations, further information on the

eigenfunction expansion can be obtained from the theory of multiparameter eigenvalue problems. This

connection will be further discussed in Remark 6.6 below.

In particular, the Fourier decomposition (2.6), combined with the eigenfunction expansion of the

Sturm-Liouville operator −∆k, gives rise to an eigenfunction expansion of (∆N ,DN ) in terms of the

separable solutions wk,λ defined in Lemma 2.1:

Proposition 2.3. There exists a sequence of locally finite positive Borel measures ρk on R+0 such that

the map h 7→ Fh, where

(Fh)k(λ) :=

∫

M

h(ξ)w−k,λ(ξ)ω(dξ)(k ∈ Z, λ ≥ 0

), (2.11)

is an isometric isomorphism F : L2(M) −→ ⊕k∈Z

L2(R+0 ,ρk) whose inverse is given by

(F−1{ϕk})(ξ) =∑

k∈Z

∫

R+0

ϕk(λ)wk,λ(ξ)ρk(dλ). (2.12)

The convergence of the integral in (2.11) is understood with respect to the norm of L2(R+0 ,ρk), and the

convergence of the inner integrals and the series in (2.12) is understood with respect to the norm of

L2(M). Moreover, the operator F is a spectral representation of (∆N ,DN ) in the sense that

DN =

{h ∈ L2(M)

∣∣∣∣∑

k∈Z

∫

R+0

λ2∣∣(Fh)k(λ)

∣∣2ρk(dλ) <∞}

(2.13)

(F(−∆Nh)

)k(λ) = λ ·(Fh)k(λ), h ∈ DN , k ∈ Z. (2.14)

Proof. Let h ∈ L2(M). By Fubini’s theorem, h(x, ·) ∈ L2(T) for a.e. x ∈ R+. For these points x we have

h(x, θ) =∑

k∈Z

hk(x) ei2kπθ , where hk(x) :=

∫ 1

0

e−i2kπϑh(x, ϑ)dϑ, (2.15)

the series converging in the norm of L2(T). It is straightforward that hk ∈ L2(A) for all k ∈ Z, and

therefore the function hk can be represented in terms of the eigenfunction expansion of the Sturm-Liouville

operator ∆k (Proposition A.2): denoting the spectral measure of ∆k by ρk, we have

hk(x) =

∫

R+0

(F∆k

hk)(λ) vk,λ(x)ρk(dλ), where

(F∆k

hk)(λ) :=

∫ ∞

0

hk(y) vk,λ(y)A(y)dy

the integrals converging in the norms of L2(A) and L2(ρk) ≡ L2(R+0 ,ρk) respectively.

By definition of hk, we have(F∆k

hk)(λ) =

∫M h(ξ)w−k,λ(ξ)ω(dξ) ≡ (Fh)k(λ), with equality in

the L2(ρk)-sense. Furthermore, by a dominated convergence argument it is clear that hk(x)ei2kπθ =∫

R+0

(F∆k

hk)(λ)wk,λ(ξ)ρk(dλ) with equality in the L2(M)-sense; therefore,

h(x, θ) =∑

k∈Z

∫

R+0

(F∆k

hk)(λ)wk,λ(ξ)ρk(dλ)

proving the inversion formula (2.11)–(2.12). Finally, the fact that the integral operator F is isometric

follows from the identities

‖h‖2L2(M) =∑

k∈Z

∥∥hk∥∥2L2(A)

=∑

k∈Z

∥∥F∆khk

∥∥2L2(ρk)

=∥∥{(Fh)k}

∥∥2⊕L2(ρk)

where the first and second steps follow from the isometric properties of the classical Fourier series and

the eigenfunction expansion of ∆k, respectively.

6

It only remains to justify the identities (2.13)–(2.14). Using (2.6) we obtain

(F(−∆Nh)

)k(λ) =

(F∆k

(−∆Nh)k)(λ) =

(F∆k

(−∆k hk))(λ) = λ ·

(Fh

)k(λ), h ∈ DN

which proves (2.14). Now, let h ∈ L2(M) be such that∑k∈Z

∫R

+0λ2

∣∣(Fh)k(λ)∣∣2ρk(dλ) < ∞. We know

that F∆kis a spectral representation of ∆k (Proposition A.2), which means in particular that

D(∆k) =

{u ∈ L2(A)

∣∣∣∣∫

R+0

λ2|(F∆ku)(λ)|2ρk(dλ) <∞

}.

Consequently, we have (hk)k∈Z ∈ ⊕k∈Z

D(∆k) and h =∑

k∈Zei2kπθ hk ∈ DN . Conversely, if h ∈ DN

then by (2.14) we have {λ ·(Fh)k(λ)} ∈⊕

k∈ZL2(ρk), and we conclude that (2.13) holds.

Since ∆N is a negative self-adjoint operator, it is the infinitesimal generator of a strongly continu-

ous semigroup in L2(M), denoted by{et∆N

}t≥0

. For any real-valued u ∈ H1(M), |u| ∈ H1(M) and

E(|u|, |u|) ≤ E(u, u). Therefore, et∆N is positivity-preserving for every t ≥ 0. Further, |u| ∧ 1 ∈ H1(M)

and E (|u| ∧ 1, |u| ∧ 1) ≤ E (|u|, |u|). Thus, for every p ∈ [1,+∞] the subspace L2(M) ∩ Lp(M) is invari-

ant under et∆N , for every t ≥ 0. The semigroup{et∆N

}t≥0

can be extended into a strongly continuous

contraction semigroup in Lp(M) (see e.g. [14, Sections 1.3–1.4]). In other words, et∆N is a strongly

continuous Markov semigroup in Lp(M) for every p ∈ [1,∞]. The analogous statement holds for the

semigroup{et∆k

}t≥0

in Lp(A), for every k ∈ Z.

Proposition 2.4. Assume that the action of et∆N on L2(M) is given by a symmetric heat kernel, i.e.

there exists a measurable function p : R+ ×M ×M −→ R+0 such that:

I. For all t, s > 0 and ξ1, ξ2 ∈M ,

p(t, ξ1, ξ2) = p(t, ξ2, ξ1) and p(t+ s, ξ1, ξ2) =

∫

M

p(t, ξ1, ξ3) p(s, ξ3, ξ2)ω(dξ3);

II. For t > 0, h ∈ L2(M) and ω-a.e. ξ1 ∈M ,

(et∆Nh)(ξ1) =

∫

M

h(ξ2) p(t, ξ1, ξ2)ω(dξ2).

Then, for t > 0 and ω-a.e. ξ1, ξ2 ∈M , the heat kernel admits the spectral representation

p(t, ξ1, ξ2) =∑

k∈Z

∫

R+0

e−tλwk,λ(ξ1)w−k,λ(ξ2)ρk(dλ) (2.16)

where the integral and the sum are absolutely convergent.

Proof. Fix t > 0. It follows from condition I that∫

M

p(t, ξ1, ξ2)2 ω(dξ2) = p(2t, ξ1, ξ1) <∞ (ξ1 ∈M),

meaning in particular that p(t, ξ1, ·) ∈ L2(M) for all ξ1 ∈ M . Moreover, by the spectral representation

property (2.13)–(2.14) we have [F(et∆Nh)]k(λ) = e−tλ(Fh)k(λ) for all h ∈ L2(M), hence

∑

k∈Z

∫

R+0

(Fh)k(λ) [Fp(t, ξ1, ·)]k(λ)ρk(dλ) = (et∆Nh)(ξ1)

= F−1

{e−t ·(Fh)k(·)

}(ξ1) =

∑

k∈Z

∫

R+0

e−tλ(Fh)k(λ)wk,λ(ξ1)ρk(dλ)

(2.17)

for ω-a.e. ξ1 ∈ M . Since h ∈ L2(M) is arbitrary, from (2.17) we deduce that {e−tλwk,λ(ξ1)} =

{[Fp(t, ξ1, ·)]k(λ)} ∈ ⊕k∈Z

L2(ρk) for ω-a.e. ξ1 ∈M . Therefore

∑

k∈Z

∫

R+0

e−tλ|wk,λ(ξ1)|2 ρk(dλ) =∑

k∈Z

∥∥e−tλ/2wk,λ(ξ1)∥∥2L2(ρk)

<∞

7

and it follows (by the Cauchy-Schwarz inequality) that the right-hand side of (2.16) is absolutely conver-

gent for ω-a.e. ξ1, ξ2 ∈M . Moreover, the isometric property of F yields

p(t, ξ1, ξ2) =⟨p(t/2, ξ1, ·), p(t/2, ξ2, ·)

⟩L2(M)

=∑

k∈Z

⟨e−tλ/2wk,λ(ξ1), e

−tλ/2wk,λ(ξ2)⟩L2(ρk)

and therefore the identity (2.16) holds for ω-a.e. ξ1, ξ2 ∈M .

Corollary 2.5. If the assumptions of Proposition 2.4 are satisfied, then for t ≥ 0, k ∈ Z, λ ∈ supp(ρk)

and ω-a.e. ξ1 ∈M we have

e−tλwk,λ(ξ1) =

∫

M

wk,λ(ξ2) p(t, ξ1, ξ2)ω(dξ2).

Proof. Fix t ≥ 0 and k ∈ Z. Notice that

p(t, ξ1, ξ2) =∑

j∈Z

ei2jπ(θ1−θ2)p∆j(t, x1, x2),

where p∆j(t, x1, x2) =

∫R

+0e−tλ vj,λ(x1) vj,λ(x2)ρj(dλ) is the heat kernel for the semigroup {et∆j} on

L2(R+, A(x)dx) (Proposition A.3), and the sum converges absolutely. Hence for λ ∈ supp(ρk) and ω-a.e.

ξ1 ∈M we can write∫

M

wk,λ(ξ2) p(t, ξ1, ξ2)ω(dξ2)

=

∫

M

ei2kπθ2vk,λ(x2)∑

j∈Z

ei2jπ(θ1−θ2)p∆j(t, x1, x2)A(x2)dx2dθ2

=

∫ ∞

0

vk,λ(x2)∑

j∈Z

ei2jπθ1∫ 1

0

ei2(k−j)πθ2dθ2 p∆j(t, x1, x2)A(x2)dx2

= ei2kπθ1∫ ∞

0

vk,λ(x2) p∆k(t, x1, x2)A(x2)dx2

= ei2kπθ1ζk(x1)

∫ ∞

0

vk,λ(x2)

∫

R+0

e−tλ0 vk,λ0(x1) vk,λ0 (x2)ρk(dλ0)Bk(x2)dx2

= ei2kπθ1e−tλvk,λ(x1)

= e−tλwk,λ(ξ1).

The second to last equality follows from the eigenfunction expansion of the Sturm-Liouville operator ℓkdefined in Lemma 2.2, considering that the double integral can be recognized as Fℓk [F−1

ℓke−t· vk,·(x1)](λ),

where (Fℓkg)(λ) :=∫R+ g(y) vk,λ(y)Bk(y)dy ≡ (F∆k

(ζk ·g))(λ). It should also be noted that

e−tλ ∈ L2(ρk), F−1ℓke−t· ∈ L1(R+, Bk(x)dx), vk,λ ∈ Cb(R

+0 )

(cf. Proposition A.3 and Lemma A.4(a)), and therefore the second to last equality, which holds initially

for ρk-a.e. λ ≥ 0, can be extended by continuity to all λ ∈ supp(ρk).

Remark 2.6. The two results above depend on the assumption that the heat kernel exists. In the general

framework of metric measure spaces, the existence of the heat kernel for the strongly continuous Marko-

vian semigroup {etL}t≥0 determined by a given sesquilinear form is equivalent to the ultracontractivity

property ‖etLh‖∞ ≤ γ(t)‖h‖L1, where γ is a positive left-continuous function on R+ (see [2, Theorem

3.1]). Besides this, there is an extensive body of work on geometric conditions which ensure the existence

of the heat kernel. For instance, it is well-known that a heat kernel exists if the generator L is the

Dirichlet or Neumann Laplacian on a smooth bounded Euclidean domain [14, 3]. We refer to [22] and

references therein for a discussion of this problem in the context of metric measure spaces.

Here we are interested in the case of the Neumann Laplacian on (M, g), L = (∆N ,DN ). We will not

discuss the general case introduced in (2.1)–(2.2), but we note that the heat kernel exists provided that

the metric g is smooth and nondegenerate:

8

Let M = R+ × T. If A belongs to C∞(R+0 ) and A(0) > 0, then there exists a heat kernel p(t, x, y) ∈

C∞(R+ × M × M) satisfying the assumptions of Proposition 2.4.

Indeed, the stated assumption on the function A ensures that we can regard (M, g) as a submanifold

of the complete smooth Riemannian manifold (R × T, g), where g = dx2 + A(x)2dθ2 and A ∈ C∞(R) is

a positive extension of the function A. Therefore, the above claim is a consequence of the fact that the

Laplace-Beltrami operator with Neumann boundary conditions on a domain of a complete Riemannian

manifold admits a heat kernel [12, Theorem 1.1].

3 Product formulas and convolutions

In this section, we establish the product formula for the eigenfunctions wk,λ, define the associated family

of convolutions, and describe the main continuity and mapping properties of the convolution structure.

Following the results in Section 2, the contents of this section rely heavily on properties of the Sturm-

Liouville operators ℓk, studied in previous works [43, 44]. For convenience, the main results used in the

current paper are presented in appendix and proofs in this section refer to them when necessary.

We start with the following product formula.

Proposition 3.1 (Product formula for wk,λ). For each k ∈ N0 and ξ1, ξ2 ∈ M there exists a positive

measure γk,ξ1,ξ2on M such that the product wk,λ(ξ1)wk,λ(ξ2) admits the integral representation

wk,λ(ξ1)wk,λ(ξ2) =

∫

M

wk,λ(ξ3)γk,ξ1,ξ2(dξ3), ξ1, ξ2 ∈M, λ ∈ C. (3.1)

Since w−k,λ = wk,λ, this result trivially extends to all k ∈ Z.

Proof. Fix k ∈ N0. Recall that wk,λ(x, θ) = ei2kπθζk(x) vk,λ(x), where vk,λ is a solution of (2.9). We saw

in Lemma 2.2 that the operator ℓk satisfies assumption (A.6), hence we can apply the existence theorem

for Sturm-Liouville type product formulas (Theorem A.5) and conclude that there exists a family of

measures {π[k]x1,x2}x1,x2∈R

+0⊂ P(R+

0 ) with supp(π[k]x1,x2) ⊂ [|x1 − x2|, x1 + x2] and such that

vk,λ(x1) vk,λ(x2) =

∫

R+0

vk,λ dπ[k]x1,x2

(x1, x2 ∈ R+0 , λ ∈ C).

Consequently, the product formula (3.1) holds for the positive measures γk,ξ1,ξ2defined by

γk,ξ1,ξ2(dξ3) =

ζk(x1)ζk(x2)

ζk(x3)νk,ξ1,ξ2

(dξ3) (3.2)

where νk,ξ1,ξ2:= π

[k]x1,x2 ⊗ δθ1+θ2 .

It follows at once from the definition that the measures νk,ξ1,ξ2are probability measures on M . Let

us introduce the convolution operators determined by these measures:

Definition 3.2. Let k ∈ Z and µ, ν ∈ MC(M). The measure

(µ ∗kν)(·) =

∫

M

∫

M

νk,ξ1,ξ2(·)µ(dξ1) ν(dξ2)

is called the ∆k-convolution of the measures µ and ν.

It is easy to check that the ∆k-convolution has the following property:

Proposition 3.3. The space (MC(M), ∗k), equipped with the total variation norm ‖µ‖ = |µ|(M), is a

commutative Banach algebra over C whose identity element is the Dirac measure δ(0,0). Moreover, the

subset P(M) is closed under the ∆k-convolution.

9

Another fundamental property of the ∆k-convolution is its connection with the ∆-Fourier transform

defined as follows:

Definition 3.4. Let µ ∈ MC(M). The ∆-Fourier transform of the measure µ is the function defined by

the integral

(Fµ)(k, λ) =

∫

M

w−k,λ(ξ)

ζk(x)µ(dξ), k ∈ Z, λ ≥ 0.

It follows from Lemma A.4(a) that ‖w−k,λ

ζk‖∞ ≤ 1, hence (Fµ)(k, λ) is well-defined for all µ ∈ MC(M)

and (k, λ) ∈ Z× R+0 .

Proposition 3.5. Let µ, ν ∈ MC(M). We have

(F(µ ∗

kν))(k, λ) = (Fµ)(k, λ) ·(Fν)(k, λ) for all k ∈ Z and λ ≥ 0. (3.3)

Moreover, for fixed k ∈ Z we have

(Fα)(k, ·) = (Fµ)(k, ·) ·(Fν)(k, ·) if and only if αk = µk ⋄kνk

where τk (τ = α, µ, ν) is the complex measure on R+0 defined by

τk(J) =

∫

M

e−i2kπθ 1J(x) τ(dξ)

and µk ⋄kνk(·) :=

∫R

+0

∫R

+0π[k]x1,x2(·) µk(dx1) νk(dx2) (here π

[k]x1,x2 ∈ P(R+

0 ) are the measures from the proof

of Proposition 3.1).

Proof. Applying the product formula (3.1), we obtain

(F(µ ∗

kν))(k, λ) =

∫

M

w−k,λ(ξ)

ζk(x)(µ ∗

kν)(dξ)

=

∫

M

∫

M

∫

M

w−k,λ(ξ3)

ζk(x3)(δξ1

∗kδξ2

)(dξ3)µ(dξ1)ν(dξ2)

=

∫

M

∫

M

w−k,λ(ξ1)

ζk(x1)

w−k,λ(ξ2)

ζk(x2)µ(dξ1)ν(dξ2) = (Fµ)(k, λ) ·(Fν)(k, λ)

so that (3.3) holds. Since ⋄k

is the convolution on R+0 associated with the Sturm-Liouville operator ℓk =

− 1Bk

ddx (Bk

ddx), the second statement is a consequence of the corresponding property of one-dimensional

generalized convolutions (Proposition A.7(a)).

Next we summarize some useful properties of the ∆-Fourier transform.

Proposition 3.6. The ∆-Fourier transform Fµ of µ ∈ MC(M) has the following properties:

(i) For each k ∈ Z, (Fµ)(k, ·) is continuous on R+0 . Moreover, if a family of measures {µj} ⊂ MC(M)

is tight and uniformly bounded, then {(Fµj)(k, ·)} is equicontinuous on R+0 .

(ii) Each measure µ ∈ MC(M) is uniquely determined by Fµ.

(iii) If {µn} is a sequence of measures belonging to M+(M), µ ∈ M+(M), and µnw−→ µ, then for each

k ∈ Z we have

(Fµn)(k, ·) −−−−→n→∞

(Fµ)(k, ·) uniformly on compact sets.

(iv) Suppose that limx→∞A(x) = ∞. If {µn} is a sequence of measures belonging to M+(M) whose

∆-Fourier transforms are such that

(Fµn)(k, λ) −−−−→n→∞

f(k, λ) pointwise in (k, λ) ∈ Z× R+0

for some real-valued function f such that f(0, ·) is continuous at a neighbourhood of zero, then

µnw−→ µ for some measure µ ∈ M+(M) such that Fµ ≡ f .

10

Proof. (i) We have

(Fµ)(k, λ) =

∫

R+0

vk,λ(x) µk(dx) ≡ (Fℓk µk)(λ)

where

(Fℓkν)(λ) =∫

R+0

vk,λ(x) ν(dx), ν ∈ P(R+0 )

is the generalized Fourier transform of measures determined by the Sturm-Liouville operator ℓk, as defined

in (A.7). Therefore, the result follows from the corresponding property of one-dimensional convolutions

(Proposition A.6(i)).

(ii) Let µ ∈ MC(M) be such that (Fµ)(k, λ) = 0 for all k ∈ Z and λ ≥ 0. Let f ∈ Cc(R+0 ) and

g ∈ C1(T). Recalling that the Fourier series g(θ) =∑

k∈Z〈g, e−i2kπ·〉 ei2kπθ converges absolutely and

uniformly [15, Theorem 1.4.2], we get

∫

M

f(x) g(θ)µ(d(x, θ)) =

∫

M

f(x)∑

k∈Z

〈g, e−i2kπ·〉 ei2kπθ µ(d(x, θ))

=∑

k∈Z

〈g, e−i2kπ·〉∫

R+0

f(x) µ−k(dx)

= 0

where the last equality holds because, by Proposition A.6(ii), (Fµ)(k, ·) ≡ 0 implies that µk = 0. By the

Stone-Weierstrass theorem (see [40, Section 38] and also [26, Corollary 15.3]), this implies that µ is the

zero measure.

(iii) This follows directly from Proposition A.6(iii).

(iv) Let us show that {µn} is tight. Fix ε > 0. Given that f(0, ·) is continuous near zero, we can

choose δ > 0 such that ∣∣∣∣1

δ

∫ 2δ

0

(f(0, 0)− f(0, λ)

)dλ

∣∣∣∣ < ε.

Furthermore, according to Lemma A.4(b) we have limx→∞ v0,λ(x) = 0 for all λ > 0; we can therefore

pick 0 < β <∞ such that

∫ 2δ

0

(1− w0,λ(ξ)

)dλ ≥ δ for all (x, θ) ∈ (β,∞)× T.

We now compute

µn([β,∞)× T) ≤ 1

δ

∫

[β,∞)×T

∫ 2δ

0

(1− w0,λ(ξ)

)dλµn(dξ)

≤ 1

δ

∫

[a,∞)×T

∫ 2δ

0

(1− w0,λ(ξ)

)dλµn(dξ)

=1

δ

∫ 2δ

0

((Fµn)(0, 0)− (Fµn)(0, λ)

)dλ

so that, using dominated convergence, we obtain

lim supn→∞

µn([β,∞)× T) ≤ 1

δlim supn→∞

∫ 2δ

0

((Fµn)(0, 0)− (Fµn)(0, λ)

)dλ =

1

δ

∫ 2δ

0

(f(0, 0)− f(0, λ)

)dλ < ε

where ε is arbitrary, showing that {µn} is tight.

Since {µn} is also uniformly bounded, Prohorov’s theorem ensures that given a subsequence {µnk},

there exists a further subsequence {µnkj} and a measure µ ∈ M+(M) for which we have µnkj

w−→ µ. By

part (iii) we must have (Fµ)(k, λ) = f(k, λ) for all (k, λ) ∈ Z×R+0 . It then follows from (ii) that all such

subsequences (and hence the sequence {µn} itself) converge weakly to a unique measure µ.

11

In the sequel we will always assume that limx→∞ A(x) = ∞.

Corollary 3.7. For each k ∈ Z, the mapping (µ, ν) 7→ µ ∗kν is continuous in the weak topology.

Proof. We have

(F(δξ1

∗kδξ2

))(j, λ) =

∫

M

e−i2jπθ3vj,λ(x3)

ζj(x3)(δξ1

∗kδξ2

)(dξ3) = e−i2jπ(θ1+θ2)∫

R+0

vj,λ(x3)

ζj(x3)(δx1 ⋄k δx2)(dx3).

(3.4)

Proposition A.7(b) ensures that (x1, x2) 7→ δx1 ⋄k δx2 is continuous in the weak topology, hence the

expression in the right hand side is a continuous function of (ξ1, ξ2). It then follows from Proposition

3.6(iv) that (ξ1, ξ2) 7→ δξ1∗kδξ2

is continuous in the weak topology.

Let h ∈ Cb(M) and µn, νn ∈ MC(M) with µnw−→ µ and νn

w−→ ν. We have just seen that∫Mh(x3) (δξ1

∗kδξ2

)(dξ3) is a continuous function of (ξ1, ξ2); consequently,

limn

∫

M

∫

M

(∫

M

h(ξ3) (δξ1∗kδξ2

)(dξ3)

)µn(dx)νn(dy) =

∫

M

∫

M

(∫

M

h(ξ3) (δξ1∗kδξ2

)(dξ3)

)µ(dx)ν(dy)

which means (since h is arbitrary) that µn ∗kνn → µ ∗

kν.

The operator T µk defined by the integral

(T µkh)(ξ) :=

∫

M

h d(δξ ∗kµ)

is said to be the ∆k-translation by the measure µ ∈ MC(M). The next result summarizes its mapping

properties. For brevity, we write Lpk := Lp(M,Bk(x)dxdθ).

Proposition 3.8. (a) If h ∈ Cb(M), then Tµkh ∈ Cb(M) for all µ ∈ MC(M).

(b) If h ∈ C0(M), then Tµkh ∈ C0(M) for all µ ∈ MC(M).

(c) Let 1 ≤ p ≤ ∞, µ ∈ M+(M) and h ∈ Lpk. The ∆k-translation (T µkh)(x) is a Borel measurable

function of x ∈M , and we have

‖T µkh‖Lp

k≤ ‖µ‖·‖h‖Lp

k. (3.5)

(d) Let p1, p2 ∈ [1,∞] such that 1p1

+ 1p2

≥ 1, and write Tξk := T

δξk (ξ ∈M). For h ∈ Lp1k and g ∈ Lp2k ,

the ∆k-convolution

(h ∗kg)(ξ) =

∫

M

(T ξ1

k h)(ξ) g(ξ1)Bk(x1)dx1dθ1

is well-defined and, for s ∈ [1,∞] defined by 1s = 1

p1+ 1

p2− 1, it satisfies

‖h ∗kg‖Ls

k≤ ‖h‖Lp1

k‖g‖Lp2

k

(in particular, h ∗kg ∈ Lsk).

Proof. (a) This is an immediate consequence of Corollary 3.7.

(b) It follows from (3.4) that for all j ∈ Z and λ > 0 we have

(F(δξ ∗

kµ))(j, λ) =

∫

M

e−i2jπ(θ+θ1)∫

R+0

vj,λ(x3)

ζj(x3)(δx ⋄

kδx1)(dx3)µ(dξ1) −→ 0 as x→ ∞ (3.6)

where the last step follows from dominated convergence and the fact that δx ⋄kδx1

v−→ 0 as x→ ∞, where

0 denotes the zero measure (Proposition A.7(d)). It follows from (3.6) and similar reasoning as in [43,

Remark 4.7] that δξ ∗kµ

v−→ 0 as x→ ∞, so that (b) holds.

12

(c) In the case p = ∞, the proof is straightforward. Let 1 ≤ p < ∞. Suppose first that h(x, θ) =

f(x)g(θ) and observe that

(T(x2,θ2)k h)(x1, θ1) = (T x2

ℓkf)(x1) ·(T θ2

Tg)(θ1)

where T xℓk

is the generalized translation associated with the Sturm-Liouville operator ℓk and (T θ2Tg)(θ1) :=

g(θ1 + θ2) is the ordinary translation on the torus. We have ‖T xℓkf‖Lp(R+, Bk(x)dx) ≤ ‖f‖Lp(R+, Bk(x)dx)

(cf. Proposition A.7(e)), and therefore

‖T (x2,θ2)k h‖Lp

k= ‖T x

ℓkf‖Lp(R+, Bk(x)dx)‖T θ2Tg‖Lp(T) ≤ ‖f‖Lp(R+, Bk(x)dx)‖g‖Lp(T) = ‖h‖Lp

k.

Since the linear span of indicator functions of compact rectangles I × J ⊂ R+0 × T is dense in Lpk, we

have ‖T (x,θ)k h‖Lp

k≤ ‖h‖Lp

kfor all h ∈ Lpk and (x, θ) ∈ M , showing that (3.5) holds for Dirac measures

µ = δ(x,θ). The result can be extended to all µ ∈ M+(M) (and 1 ≤ p <∞) by using Minkowski’s integral

inequality.

(d) The proof relies on part (c) and the same reasoning as in the classical case; see e.g. the proof of

Proposition 1.III.5 of [47].

In the next statement we show that if a heat kernel exists for the heat semigroup {et∆N}, then the

functions et∆Nwk,λ = e−tλwk,λ also admit a product formula whose measures do not depend on the

spectral parameter λ and, moreover, are absolutely continuous with respect to ω.

Proposition 3.9. Assume that the action of et∆N on L2(M) is given by a symmetric heat kernel satisfying

conditions I and II of Proposition 2.4. Let γt,k,ξ1,ξ2be the positive measure defined by

γt,k,ξ1,ξ2(dξ3) =

∫

M

γk,ξ4,ξ2(dξ3) p(t, ξ1, ξ4)ω(dξ4).

Then, the product e−tλwk,λ(ξ1)wk,λ(ξ2) admits the integral representation

e−tλwk,λ(ξ1)wk,λ(ξ2) =

∫

M

wk,λ(ξ3)γt,k,ξ1,ξ2(dξ3) (t ≥ 0, ξ1, ξ2 ∈M, λ ∈ supp(ρk)).

Proof. By direct calculation we get∫

M

wk,λ(ξ3)γt,k,ξ1,ξ2(dξ3) =

∫

M

∫

M

wk,λ(ξ3)γk,ξ4,ξ2(dξ3) p(t, ξ1, ξ4)ω(dξ4)

= e−tλwk,λ(ξ1)wk,λ(ξ2)

where, by Proposition 3.1 and Corollary 2.5, the last equality holds for t ≥ 0, λ ∈ supp(ρk), ξ2 ∈M and

ω-a.e. ξ1 ∈ M . Using the symmetry relation∫Mwk,λ(ξ3)γt,k,ξ1,ξ2

(dξ3) =∫Mwk,λ(ξ3)γt,k,ξ2,ξ1

(dξ3),

the identity extends by continuity to all ξ1, ξ2 ∈ M . (The given symmetry can be deduced by noting

that, by Propositions 2.3–2.4 and Proposition A.7(c), we have for g ∈ C2c(R

+)∫

M

ei2kπθ3g(x3)γt,k,ξ1,ξ2(dξ3) =

∫

M

∫

R+0

wk,λ(ξ4)wk,λ(ξ2) (F∆kg)(λ)ρk(dλ) p(t, ξ1, ξ4)ω(dξ4)

=

∫

R+0

e−tλwk,λ(ξ1)wk,λ(ξ2) (F∆kg)(λ)ρk(dλ)

and, therefore, ( γt,k,ξ1,ξ2)−k = ( γt,k,ξ2,ξ1

)−k.)

4 Infinitely divisible measures and convolution semigroups

In this section we develop the basic notions of divisibility of measures with respect to the convolution

algebras (MC(M), ∗k). As in the classical theory, these will be seen to induce a Lévy-Khintchine type

representation and to a convolution semigroup representation for the reflected Brownian motion on (M, g).

First we present the following basic definitions:

13

Definition 4.1.

• The set Pk,id of ∆k-infinitely divisible measures is defined by

Pk,id ={µ ∈ P(M)

∣∣ for all n ∈ N there exists νn ∈ P(M) such that µ = (νn)∗kn}

(4.1)

where (νn)∗kn denotes the n-fold ∆k-convolution of νn with itself.

• The ∆k-Poisson measure associated with ν ∈ M+(M) is

ek(ν) := e−‖ν‖∞∑

n=0

ν∗kn

n!

(the infinite sum converging in the weak topology).

• A measure µ ∈ Pk,id is called a ∆k-Gaussian measure if the measures νn in (4.1) are such that

limn→∞

n ·νn(M \ V ) = 0 for every open set V containing (0, 0).

It is easy to check that, for ν ∈ M+(M),

∫

M

e−i2jπθ vk,λ(x) ek(ν)(dξ) = exp

(∫

M

[e−i2jπθ vk,λ(x)− 1

]ν(dξ)

), (j, λ) ∈ Z× R

+0 . (4.2)

(This is an equivalent characterization of ∆k-Poisson measures, because by [5, Theorem 2.2.4] each

measure µ ∈ MC(M) is characterized by the integrals∫Me−i2jπθ vk,λ(x)µ(dξ).) More generally, if the

positive measure ν is (possibly) unbounded and the equality (4.2) holds for some measure ek(ν) ∈ P(M),

then we will also say that ek(ν) is a ∆k-Poisson measure associated with ν.

Definition 4.2. A family {µt}t≥0 ⊂ P(M) is called a ∆k-convolution semigroup if it satisfies the condi-

tions

µs ∗kµt = µs+t for all s, t ≥ 0, µ0 = δ(0,0) and µt

w−→ δ(0,0) as t ↓ 0.

The ∆k-convolution semigroup {µt}t≥0 is said to be Gaussian if µ1 is a ∆k-Gaussian measure.

A measure µ ∈ MC(M) is said to be symmetric if µ(B) = µ(B) for all Borel subsets B ⊂M , where B

is the image of B under the mapping (x, θ) 7→ (x, 1− θ). One can show that for each symmetric measure

µ ∈ Pk,id there exists a unique ∆k-convolution semigroup {µt}t≥0 such that µ1 = µ; consequently,

there is a one-to-one correspondence between symmetric ∆k-infinitely divisible measures and symmetric

∆k-convolution semigroups. (The proof is similar to that of the corresponding result for the ordinary

convolution on the torus, see also [5, Theorem 5.3.4].)

It follows from Proposition A.8 that the convolution algebra (M, ∗k) is a product hypergroup in the

sense of [5, Definition 1.5.29]. We can therefore use a general result on infinitely divisible measures

on commutative hypergroups [37, Theorems 4.4 and 4.7] to obtain the following Lévy-Khintchine type

representation for symmetric ∆k-infinitely divisible measures (and for the corresponding convolution

semigroups):

Proposition 4.3. Any symmetric measure µ ∈ Pk,id can be represented as

µ = γ ∗kek(ν)

where ek(ν) is the ∆k-Poisson measure associated with the σ-finite positive measure ν = limt↓0

(1tµt)|M\(0,0)and γ is a ∆k-Gaussian measure.

The representation is unique, i.e. if µ = γ ∗kek(ν) for a σ-finite positive measure ν and a Gaussian

measure γ, then ν = ν and γ = γ.

It is easy to show (cf. [38, Proposition 2.1]) that each ∆k-convolution semigroup gives rise to a

Markovian contraction semigroup of operators:

14

Proposition 4.4. Let {µt} be a ∆k-convolution semigroup. Then

(Tth)(ξ) := (T µt

k h)(ξ) =

∫

M

h d(δξ ∗kµt)

defines a conservative Feller semigroup on C0(M) such that the identity TtTνkf = T

νkTtf holds for all

t ≥ 0 and ν ∈ MC(M). The restriction{Tt|Cc(M)

}can be extended to a strongly continuous contraction

semigroup {T (p)t } on the space Lp(M) (1 ≤ p < ∞). Moreover, the operators T

(p)t are given by T

(p)t f =

Tµt

k f (f ∈ Lp(M)).

Next we show that the heat semigroup generated by ∆N is of the convolution semigroup type, in the

sense that its action can be represented in terms of integrals with respect to Gaussian ∆k-convolution

semigroups:

Proposition 4.5. For k ∈ Z, let m0 ∈ MC(M) be an absolutely continuous measure with respect to ω

whose density function qm0 belongs to L2(M) ∩ L1(M, ζk ·ω), and such that (m0)j = 0 for each j 6= k.

Then there exists a Gaussian ∆k-convolution semigroup {µkt }t≥0 such that

∫

M

(et∆Nh)(ξ)m0(dξ) =

∫

M

h(ξ)

ζk(x)

(µkt ∗k (ζk ·m0)

)(dξ)

(h ∈ L2(M), t ≥ 0

). (4.3)

Proof. For t > 0, let µkt = αkt ⊗ δ0, where {αkt }t≥0 is the ⋄k

-Gaussian convolution semigroup generated

by ℓk (Proposition A.9(a)). We recall from the proof of Corollary 2.5 that we have e−tλ ∈ L2(ρk) and

αkt (dx) = (F−1ℓke−t·)(x)Bk(x)dx, where F−1

ℓke−t· ∈ L1(R+, Bk(x)dx).

Our first claim is that the measure 1ζk(µkt ∗k (ζk ·m0)) is absolutely continuous with respect to ω and

that its density function qµkt ,m0

belongs to L2(M). Note first that, by assumption, (m0)j = 0 for j 6= k,

and therefore (e.g. by Proposition 3.6(ii)) m0 = (m0)k ⊗ φk, where φk is the measure on T defined by

φk(dθ) = ei2kπθdθ. We thus have

µkt ∗k (ζk ·m0) = (αkt ⋄k (ζk ·(m0)k))⊗ φk.

The absolute continuity assumption on m0 implies that (m0)k(dx) = (qm0)k(x)A(x)dx with (qm0)k ∈L2(A), so we can now use the properties of the convolution ⋄

k(see Proposition A.7(f)) to conclude that

1ζk(αkt ⋄k (ζk·(m0)k)) is also absolutely continuous with respect to A(x)dx with density belonging to L2(A),

and this proves the claim.

Let h ∈ L2(M). Combining the above with Proposition 2.3, we may now compute

∫

M

(et∆Nh)(ξ)m0(dξ) =⟨et∆Nh, qm0

⟩L2(M)

=∑

j∈Z

⟨F(et∆Nh)j , (F qm0)j

⟩L2(ρj)

=⟨e−t·(Fh)−k, (F qm0)−k

⟩L2(ρk)

=⟨(Fh)−k, (Fµ

kt )(−k, ·) (F qm0)−k

⟩L2(ρk)

=∑

j∈Z

⟨(Fh)j , (F qµk

t ,m0)j⟩L2(ρj)

=⟨h, qµk

t ,m0

⟩L2(M)

=

∫

M

h(ξ)

ζk(x)

(µkt ∗k (ζk ·m0)

)(dξ)

so that (4.3) holds.

As observed in Section 2, the sesquilinear form E associated with the heat semigroup et∆N is a

nonnegative, closed, Markovian symmetric form defined on H1(M)×H1(M); in other words, (E , H1(M))

is a Dirichlet form on L2(M). One can also check (cf. [6, 19]) that the Dirichlet form (E , H1(M))

is regular, that is, H1(M) ∩ Cc(M) is dense both in H1(M) with respect to the norm ‖u‖H1(M) =√E(u, u) + ‖u‖L2(M) and in Cc(M) with respect to the sup norm. Therefore, by a basic result from

the theory of Dirichlet forms [19, Theorem 7.2.1], there exists a Hunt process with state space M whose

transition semigroup {Pt}t≥0 is such that Ptu is, for all u ∈ Cc(M), a quasi-continuous version of et∆Nu.

15

(A Hunt process is essentially a strong Markov process whose paths are right-continuous and quasi-left-

continuous; for details we refer to [19, Appendix A.2].)

Accordingly, (4.3) can be rewritten as

Em0 [h(Wt)] =

∫

M

h d(µkt ⋆km0),

(h ∈ L2(M), t ≥ 0

)(4.4)

where:

• {Wt}t≥0 is the reflected Brownian motion on the manifold (M, g), i.e. {Wt} is the Hunt process on

M determined by the regular Dirichlet form (E , H1(M));

• Em0 is the expectation operator of the process with initial distribution m0 ∈ MC(M) (defined as

Em0 [h(Wt)] :=∫M

Eξ[h(Wt)]m0(dξ), where Eξ is the usual expectation operator for the process

started at the point ξ);

• The convolution ⋆k

is defined by ν1 ⋆kν2 = 1

ζk

((ζk ·ν1) ∗

k(ζk ·ν2)

)or, equivalently, by (ν1 ⋆

kν2)(·) =∫

M

∫Mγk,ξ1,ξ2

(·) ν1(dξ1) ν2(dξ2), with γk,ξ1,ξ2given as in (3.2);

• µkt :=µkt

ζk(so that µkt satisfies the convolution semigroup property with respect to ⋆

k).

Corollary 4.6. Let m0 ∈ MC(M) be an absolutely continuous measure with respect to ω whose den-

sity function qm0 belongs to L2(M) ∩(⋂∞

k=0 L1(M, ζk ·ω)

). Then there exist Gaussian ∆k-convolution

semigroups {µkt }t≥0 such that

∫

M

(et∆Nh)(ξ)m0(dξ) =∑

k∈Z

∫

M

ei2kπθhk(x)

ζk(x)

(µkt ∗k (ζk ·m0,−k)

)(dξ)

(h ∈ L2(M), t ≥ 0

)

where m0,k = (m0)k ⊗ φk and hk is given as in (2.15).

Proof. We have

∫

M

et∆N

(∑

k∈Z

ei2kπθ hk(x)

)m0(dξ) =

∑

k∈Z

∫

M

et∆N(ei2kπθ hk(x)

) ((m0)−k ⊗ φ−k

)(dξ).

Since each measure (m0)−k ⊗ φ−k satisfies((m0)−k ⊗ φ−k

)j= 0 for j 6= −k, the corollary follows by

applying Proposition 4.5 to each term in the right-hand side.

We now extend the result of Proposition 4.5 to other Markovian semigroups whose generators are

functions (in the functional calculus sense) of the Laplace-Beltrami operator.

Proposition 4.7. For k ∈ Z, let m0 ∈ MC(M) be an absolutely continuous measure with respect to ω

whose density function qm0 belongs to L2(M) ∩ L1(M, ζk ·ω), and such that (m0)j = 0 for each j 6= k.

Let ψk be a function of the form

ψk(λ) = cλ+

∫

R+

(1 − vk,λ(x)) τ(dx) (λ ≥ 0) (4.5)

where c ≥ 0 and τ is a σ-finite measure on R+ which is finite on the complement of any neighbourhood of

0 and such that∫R+(1− vk,λ(x)) τ(dx) <∞ for λ ≥ 0. Assume also that e−tψk(·) ∈ L2(ρk) for all t > 0.

Then there exists a ∆k-convolution semigroup {µψk

t }t≥0 such that

∫

M

(e−tψk(−∆N)h)(ξ)m0(dξ) =

∫

M

h(ξ)

ζk(x)

(µψk

t ∗k(ζk ·m0)

)(dξ)

(h ∈ L2(M), t ≥ 0

)(4.6)

where e−tψk(−∆N) is defined via the spectral theorem for the self-adjoint operator (−∆N ,DN ).

We observe that, since e−tλ ∈ L2(ρk) for all t > 0, the assumption e−tψk(·) ∈ L2(ρk) is automatically

satisfied whenever c > 0 in the right hand side of (4.5).

16

Proof. By Proposition A.9(b), there exists a ⋄k

-convolution semigroup {αψk

t }t≥0 such that (Fℓk αψk

t )(λ) =

e−tψk(λ). Using Proposition A.2 and the assumption e−tψk(·) ∈ L2(ρk), we deduce that αψk

t (dx) =

(F−1ℓke−tψk(·))(x)Bk(x)dx, where F−1

ℓke−tψk(·) ∈ L1(R+, Bk(x)dx). The result can now be proved using

the same argument as in Proposition 4.5 above.

The sesquilinear form Eψk : D(Eψk ) × D(Eψk) −→ C associated with the Markovian self-adjoint

operator −ψk(−∆N ), defined as

D(Eψk) = D(√

ψk(−∆N )), Eψk(u, v) =

⟨√ψk(−∆N )u,

√ψk(−∆N ) v

⟩L2(M)

,

is a regular Dirichlet form on L2(M). (We can prove this claim using the upper bound (A.8) and the

proof of Proposition 3.1 of [32].) Accordingly, the result stated before Corollary 4.6 ensures that there

exists a Hunt process {Xt}t≥0 with state space M such that (e−tψk(−∆N )h)(ξ) = Eξ[h(Xt)], and therefore

the convolution semigroup property (4.6) translates into the Lévy-like representation

Em0 [h(Xt)] =

∫

M

h d(µψk

t ⋆km0)

(h ∈ L2(M), t ≥ 0

)

for the law of the process {Xt}. (Here m0 is any complex measure satisfying the assumptions in Propo-

sition 4.7.) The representation (4.4) for the law of reflected Brownian motion on (M, g) is a particular

case of this result.

Remark 4.8. In general we cannot state a counterpart of Corollary 4.6 for the semigroup e−tψk(−∆N ).

This would only be possible if ψk(λ) did not depend on k, i.e. if a given function ψ(λ) could be written,

for each k = 0, 1, . . ., as ckλ+∫(1− vk,λ) dτk with ck ≥ 0 and τk measures satisfying the conditions above,

but there are no reasons to expect that this is possible other than in the trivial case ψ(λ) = cλ. (See [52],

where it is shown that the stable infinitely divisible measures for the convolution ⋄k

are not the same for

different values of k.)

5 Examples

We now review some special cases in which the theory of special functions provides further information on

the eigenfunction expansion of ∆N and the associated convolution structure. We start with an example

where the solutions wk,λ can be expressed in terms of the Whittaker function of the second kind, and the

Fourier decomposition gives rise to a family of Sturm-Liouville operators which are generators of drifted

Bessel processes.

Example 5.1. Consider the case A(x) = x1/2, so that the Riemannian metric on M = R+ × T is

g = dx2 + x dθ2, the volume form is dω =√x dxdθ and the Laplace-Beltrami operator on (M, g) is

∆ = ∂2x +12x∂x +

1x∂

2θ .

(i) Let Mα,ν(z) := zνMα,ν(z), where Mα,ν(z) denotes the Whittaker function of the first kind [34,

§13.14]. The unique solution of the boundary value problem (2.7) is given by

wk,λ(x, θ) = ei2kπθM2(kπ)2i√

λ,− 1

4

(2ix√λ). (5.1)

Indeed, the function vk,λ(x) = e−πi/8(2x√λ)−1/4M2(kπ)2i

2√

λ,− 1

4

(2ix√λ) is, according to [35, Equation

2.1.2.108], a solution of −∆k v = λv, and we can use the results of [34, §13.14(iii) and §13.15(ii)] to

check that vk,λ(0) = 1 and (Av′k,λ)(0) = 0.

(ii) Let rρk(λ) := 2−3/2π−2λ−1/4 exp

(− 2k2π3

√λ

)∣∣Γ(14 − 2(kπ)2i√

λ

)∣∣2. The integral operator F : L2(M) −→⊕k∈Z

L2(R+, rρk(λ)dλ) defined by

(Fh)k(λ) :=

∫ ∞

0

∫ 1

0

h(x, θ) e−i2kπθdθM2(kπ)2i√

λ,− 1

4

(2ix√λ)x1/2dx

17

is a spectral representation of the Laplace-Beltrami operator (cf. Proposition 2.3), and its inverse

is given by

(F−1{ϕk})(ξ) =∑

k∈Z

∫ ∞

0

ϕk(λ) ei2kπθM2(kπ)2i√

λ,− 1

4

(2ix√λ) rρk

(λ)dλ.

By Proposition 2.3, to prove (ii) we only need to show that the spectral measure of the self-adjoint

realization of ∆k determined by the boundary condition (Av′)(0) = 0 is given by ρk(dλ) = rρk(λ)dλ.

But this fact is a consequence of the general results of [31] on spectral representations associated with

the Sturm-Liouville operator ℓν,µ = − d2

dx2 − (2ν+1x +µ) ddx . Indeed, it follows from [31, Proposition 1] (see

also [45]) that the integral transforms

(Qf)(τ) = (2τ)−1/4

∫ ∞

0

f(x)x1/4 e8(kπ)2x−πi/8M2(kπ)2i

τ,− 1

4

(2ixτ) dx

(Q−1η)(x) =x1/4 e−8(kπ)2x−πi/8

23/4 π2

∫ ∞

0

η(τ)M2(kπ)2 i

τ,− 1

4

(2ixτ) τ1/4 exp(−2k2π3

τ

) ∣∣∣∣Γ(1

4− 2(kπ)2i

τ

)∣∣∣∣2

dτ

define an isometric isomorphism L2(R+, x1/2e(4kπ)2x) −→ L2

(R+, ( τ2 )

1/2 π−2 exp(− 2k2π3

τ )∣∣Γ(14−

2(kπ)2iτ )

∣∣2dτ)

satisfying Q(ℓ− 14 ,8(kπ)

2f)(λ) = λ ·(Qf)(λ). Since the operators ℓ− 14 ,8(kπ)

2 and ∆k are related via an ele-

mentary change of variables, we easily conclude that ρk(dλ) = rρk(λ)dλ.

(iii) For each k ∈ N0 and ξj = (xj , θj) ∈ M (j = 1, 2) there exists a positive measure γk,ξ1,ξ2on M

such that for all τ ∈ C the generalized eigenfunctions (5.1) satisfy

ei2kπ(θ1+θ2)M2(kπ)2i

τ,− 1

4

(2ix1τ)M2(kπ)2i

τ,− 1

4

(2ix2τ) =

∫

M

ei2kπθ3M2(kπ)2i

τ,− 1

4

(2ix3τ)γk,ξ1,ξ2(dξ3).

(5.2)

The support of measure γk,ξ1,ξ2is the set [|x1 − x2|, x1 + x2]× {θ1 + θ2}.

(iii’) When k = 0, the product formula (5.2) reduces to

J− 14(x1τ)J− 1

4(x2τ) =

∫

M

J− 14(x3τ)γ0,ξ1,ξ2

(dξ3) (5.3)

where Jα(τx) := 2αΓ(α + 1)(τx)−αJα(τx) and Jα is the Bessel function of the first kind. The

measures γ0,ξ1,ξ2in (5.3) are explicitly given by

γ0,ξ1,ξ2(dξ3) =

23/2Γ(34 )√π Γ(14 )

[(x23 − (x1 − x2)

2)((x1 + x2)2 − x23)

]−3/4

×√x1x2 x31[|x1−x2|,x1+x2](x3) dx3 δθ1+θ2(dθ3).

Property (iii) is a particular case of Proposition 3.1. The identity M0,− 14(2ixτ) = J− 1

4(xτ) (cf. [34,

§10.27 and §13.18(iii)]) leads to (5.3). The closed-form expression for the measures γ0,ξ1,ξ2(dξ3) follows

from the well-known product formula for the Bessel function of the first kind [23, 50]. We mention

that the convolution (µ⋆0ν)(·) =

∫M

∫M γ0,ξ1,ξ2

(·)µ(dξ1) ν(dξ2) is (modulo the product with the trivial

convolution on the torus) a particular case of the Bessel-Kingman convolution [25], which is one of most

notable examples of Sturm-Liouville hypergroups [5, 38].

It is natural to conjecture that the measures γk,ξ1,ξ2(k = 1, 2, . . .) can also be written in closed form

in terms of classical special functions. If this is true, determining such a closed form expression is likely

to require a detailed and nontrivial analysis of the properties of the Whittaker function of the first kind.

(Compare with e.g. [28, 29, 42], where nontrivial product formulas have been determined for other special

functions.)

According to [6], one can formally interpret the manifold (M, g) as a cone-like surface of revolution

S = {(t, r(t) cos θ, r(t) sin θ) | t > 0, θ ∈ T} with profile r(t) ∼√t as t ↓ 0. The properties of self-

adjoint extensions of the Laplace-Beltrami operator (and the corresponding Markovian semigroups) on

such cone-like manifolds have been widely studied, see [6] and references therein. As a particular case of

Corollary 4.6, we obtain the following convolution semigroup property for the heat semigroup generated

by the Neumann realization of the Laplace-Beltrami operator on (M, g):

18

(iv) If m0 ∈ MC(M) satisfies the absolute continuity assumption of Corollary 4.6, then the transition

probabilities of the reflected Brownian motion {Wt} on the manifold (M, g) with initial distribution

m0 can be written as

Em0 [h(Wt)] ≡∫

M

(et∆Nh)(ξ)m0(dξ)

=∑

k∈Z

∫

M

ei2kπθ hk(x)(µkt ⋆k

m0,−k)(dξ)

(h ∈ L2(M), t ≥ 0

) (5.4)

where {µkt }t≥0 is a convolution semigroup with respect to the convolution ⋆k

defined by (µ⋆kν)(·) =∫

M

∫Mγk,ξ1,ξ2

(·)µ(dξ1) ν(dξ2), hk(x) :=∫ 1

0e−i2kπϑh(x, ϑ)dϑ and the measures m0,−k are defined

as in Corollary 4.6.

As we saw earlier, the convolution semigroups {µkt } are of the form (αk

t

ζk)⊗ δ0, where {αkt } is the law of

the one-dimensional diffusion process (started at x = 0) generated by the Sturm-Liouville operator ℓkdefined in (2.10). The differential equation ℓku = λu can be transformed, by the change of dependent

variable U(x) = ζk(x)e−2k2xu(x), into an equation of the form LkU = ΛU (Λ ∈ C), where Lk =

− d2

dx2 −(

12x+(4kπ)2

)ddx , i.e. Lk is the infinitesimal generator of a Bessel process with constant drift (4kπ)2

[31]. Therefore, the identity (5.4) shows that the transition probabilities of {Wt} can be decomposed in

terms of transition probabilities of (one-dimensional) drifted Bessel processes.

Finally, we call attention to the following convolution semigroup representation for Markovian semi-

groups generated by fractional powers of the Laplace-Beltrami operator, which can be deduced from

Proposition 4.7:

(v) Let m0 ∈ MC(M) satisfy the assumptions of Proposition 4.7 and (m0)j = 0 for each j 6= 0. Let

0 < q < 1. Then the Markovian semigroup generated by the operator −(−∆N)q is such that

∫

M

(e−t(−∆N )qh)(ξ)m0(dξ) =

∫

M

h(ξ)(νq,t ⋆

0m0

)(dξ)

(h ∈ L2(M), t ≥ 0

)

where {νq,t}t≥0 is a ⋆0-convolution semigroup.

(To prove (v), we also need to recall the following special property of the Bessel-Kingman convolution

[48, Theorem 2]: for each 0 < q < 1 there exists a measure σq ∈ P(R+0 ) which is infinitely divisible with

respect to the Bessel-Kingman convolution ⋆0

and such that∫R

+0J− 1

4(x√λ)σq(dx) = e−λ

q

. It is easy to

check that e−tλq ∈ L2(ρ0) for all t > 0.)

Example 5.2. Consider now the more general case A(x) = xβ with 0 < β < 1. The corresponding

Riemannian metric, g = dx2 + x2βdθ2, endows the space M = R+ × T with a metric structure which,

like in the previous example, can be formally interpreted as that of a surface of revolution with profile

r(t) ∼ tβ as t ↓ 0.

If β 6= 12 , the solution of the boundary value problem (2.7) and the spectral measures ρk (k = 1, 2, . . .)

can no longer be written in closed form. Nevertheless, the convolution semigroup property of the Laplace-

Beltrami operator ∆ = ∂2x+βx∂x+

1x2β ∂

2θ on the cone-like manifold (M, g), stated in property (iv) of the

previous example, continues to hold in this more general setting.

For k = 0, the solutions of −∆0v(x) ≡ −v′′(x)− βxv

′(x) = λv(x) are the normalized Bessel functions

with parameter β−12 . Therefore, we have the following extension of property (iii’) of the preceding

example: the product formula

Jβ−12(x1τ)Jβ−1

2(x2τ) =

∫

M

Jβ−12(x3τ)γ0,ξ1,ξ2

(dξ3) (τ ∈ C)

holds for all ξ1, ξ2 ∈M , where the measures γ0,ξ1,ξ2are given by

γ0,ξ1,ξ2(dξ3) =

22−βΓ(β+12 )

√π Γ(β2 )

[(x23 − (x1 − x2)

2)((x1 + x2)2 − x23)

]β/2−1

19

× (x1x2)1−β x31[|x1−x2|,x1+x2](x3) dx3 δθ1+θ2(dθ3).

The convolution semigroup representation for {e−t(−∆N)q}t≥0, formulated in property (v) of the previous

example, also extends to the case 0 < β < 1 without any essential change.

In the latter example, the limiting case β = 0 corresponds to the standard metric structure on

the cylinder R+ × T, which is a trivial case because a convolution associated with the Laplace-Beltrami

operator ∆ = ∂2x+∂2θ can be introduced in a trivial way (namely, by taking the product of one-dimensional

convolutions). In this trivial case, the convolutions introduced in the previous sections have a particularly

simple structure:

Example 5.3. If A ≡ 1, the Fourier decomposition (2.6) yields the Sturm-Liouville operators ∆k =

∂2x− (2kπ)2. The eigenfunction expansion (2.11)–(2.12) is simply a composition of a Fourier series in the

variable θ and a cosine Fourier transform in the variable x,

(Fh)k(λ) =

∫ ∞

0

∫ 1

0

h(ξ)e−i2kπθdθ cos(xλk) dx (λk =√λ− (2kπ)2)

(F−1{ϕk})(ξ) =1

π

∑

k∈Z

∫ ∞

(2kπ)2ϕk(λ)e

i2kπθ cos(xλk) λ−1k dλ,

and the product formula wk,λ(ξ1)wk,λ(ξ2) =∫Mwk,λ dγk,ξ1,ξ2

(where wk,λ(ξ1) = e−i2kπθ cos(xλk)) holds

for the measures γk,ξ1,ξ2= 1

2 (δ|x1−x2| + δx1+x2) ⊗ δθ1+θ2 , which do not depend on k. The convolution

⋆ ≡ ⋆k

is such that

δξ1⋆ δξ2

= (δx1 ⋄sym

δx2)⊗ (δθ1 ⋄Tδθ2),

i.e. it can be interpreted as a product of the convolution ⋄sym

of the symmetric hypergroup on R+0 and

the ordinary convolution ⋄T

on T (this is a product hypergroup structure, cf. [5, Definition 1.5.29 and

Example 3.5.73]). In turn, the convolution ∗k

of Definition 3.2 is such that

δξ1∗kδξ2

= (δx1 ⋄chk

δx2)⊗ (δθ1 ⋄Tδθ2),

where δx1 ⋄chk

δx2 = cosh(2kπ|x1−x2|)2 cosh(2kπx1) cosh(2kπx2)

δ|x1−x2| +cosh(2kπ(x1+x2))

2 cosh(2kπx1) cosh(2kπx2)δx1+x2 is the convolution of a

cosh hypergroup [5, Example 3.5.71]. It is interesting to note that the convolution ∗k

is a modification of

the convolution ⋆, as defined in the theory of hypergroups [5, Section 2.3]. We also observe that, since ⋆

is the convolution of the symmetric hypergroup, Corollary 4.6 specializes to the following fact: under the

assumption on m0 ∈ MC(M) given in Corollary 4.6, the transition probabilities of the reflected Brownian

motion {Wt} on (M, g) are such that

Em0 [h(Wt)] =∑

k∈Z

e−(2kπ)2t

∫

R+0

hk(x)(pR

+0 ,t

⋄sym

(m0)−k)(dx)

(h ∈ L2(M), t ≥ 0

)

where hk(x) :=∫ 1

0e−i2kπϑh(x, ϑ)dϑ, and p

R+0 ,t

is the law of a reflected Brownian motion on R+0 started

at 0, which satisfies pR

+0 ,t+s

= pR

+0 ,t

⋄sym

pR

+0 ,s

.

Next we give two other particular cases of metrics g where, as in Example 5.2, the corresponding

families of convolutions ∗k

are related with well-known one-dimensional generalized convolutions:

Example 5.4. Consider A(x) = (sinhx)2α+1(coshx)2β+1, which satisfies condition (2.2) provided that

− 12 ≤ β ≤ α < 0 with α 6= − 1

2 . The action of the Laplace-Beltrami operator (2.3) on functions which

do not depend on θ is the same as that of the Sturm-Liouville operator ∆0 = ∂2x + [(2α + 1) coth(x) +

(2β + 1) tanh(x)]∂x. The solution of the boundary value problem −∆0v = λv, v(0) = 1, (Av′)(0) = 0 is

the Jacobi function

v0,λ(x) = 2F1

(12 (σ − iτ), 12 (σ + iτ);α+ 1;−(sinhx)2

)(σ = α+ β + 1, λ = τ2 + σ2)

20

for which the product formula v0,λ(x1) v0,λ(x2) =∫∞0v0,λ dπ

[0]x1,x2 holds with

π[0]x1,x2

(dx3) =2−2σΓ(α+ 1)(coshx1 coshx2 coshx3)

α−β−1

√π Γ(α+ 1

2 )(sinh x1 sinhx2 sinhx3)2α×

× (1− Z2)α−1/22F1

(α+ β, α− β;α+ 1

2 ;12 (1− Z)

)1[|x1−x2|,x1+x2](x3)A(x3)dx3

where Z := (cosh x1)2+(coshx2)

2+(cosh x3)2−1

2 cosh x1 cosh x2 cosh x3, see [27]. The convolution determined by this product for-

mula is the so-called Jacobi convolution, which also gives rise to a notable example of a Sturm-Liouville

hypergroup [5, Example 3.5.64].

It follows from Proposition 3.1 that the generalized eigenfunctions of the Sturm-Liouville operator

∆k = ∆0 − (2kπ)2(sinhx)−4α−2(coshx)−4β−2 also admit a similar product formula, whose measures are

also supported on [|x1 − x2|, x1 + x2]. The results of the previous sections therefore show that the Jacobi

convolution naturally extends into a convolution structure associated with the Laplace-Beltrami operator

on (M, g).

Example 5.5. Consider A(x) = (1+ x)2. The first Fourier component of the Laplace-Beltrami operator

(2.3) is ∆0 = ∂2x+2

1+x∂x. The solution of the boundary value problem −∆0v = λv, v(0) = 1, (Av′)(0) = 0

is the function

v0,λ(x) =

{1

1+x [cos(τx) +1τ sin(τx)], τ > 0

1, τ = 0(λ = τ2)

and the product formula for this function is v0,λ(x1) v0,λ(x2) =∫∞0 v0,λ dπ

[0]x1,x2 , where the measures are

given by [51, Example 4.10]

π[0]x1,x2

=1

2(1 + x1)(1 + x2)

[(1 + |x1 − x2|)δx1−x2(dx3)

+ (1 + x1 + x2)δx1+x2(dx3) + (1 + x3)1[|x1−x2|,x1+x2](x3)dx3].

The convolution determined by this product formula gives rise to the so-called square hypergroup. It

follows from [51, Theorem 3.14] that all the product formula measures π[k]x1,x2 associated with the Fourier

components ∆k (cf. proof of Proposition 3.1) share with π[0]x1,x2 the property of having both a discrete

component supported on {|x1 − x2|, x1 + x2} and an absolutely continuous component supported on

[|x1 − x2|, x1 + x2].

The Riemannian metric g = dx2 + (1 + x)4dθ2 satisfies the additional assumption in Remark 2.6;

therefore, there exists a heat kernel for the Laplace-Beltrami operator ∆ = ∂2x +2

1+x∂x +1

(1+x)4 ∂2θ and,

as we saw in Proposition 3.9, our results on the existence of product formulas not depending on λ extend

to all the functions et∆Nwk,λ = e−tλwk,λ (t ≥ 0).

In all the examples above, the support of the convolution δξ1⋆kδξ2

= γk,ξ1,ξ2does not depend on the

parameter k. Our final example shows that this is not always the case:

Example 5.6. Let φ ∈ C∞c (R+

0 ) be a nonnegative decreasing function with supp(φ) = [0, S] and let

A(x) = exp(∫ x0φ(y)dy). We know that π

[k]x1,x2 = δx1 ⋄k δx2 , where ⋄

kis the convolution associated with

the Sturm-Liouville operator ℓk = − 1Bk

ddx(Bk

ddx). According to the general result of [44, Proposition 5.7]

on the support of convolutions associated with Sturm-Liouville operators, we have

supp(π[0]x1,x2

) =

{[|x1 − x2|, x1 + x2], min{x1, x2} ≤ 2S

[|x1 − x2|, 2S + |x1 − x2|] ∪ [x1 + x2 − 2S, x1 + x2] min{x1, x2} > 2S

and

supp(π[k]x1,x2

) = [|x1 − x2|, x1 + x2], k ≥ 1.

21

6 Product formulas and convolutions associated with elliptic op-

erators on subsets of R2

In the remainder of this paper, we will show that the techniques used above can also be used to construct

families of generalized convolution operators associated with elliptic differential operators on R+×I ⊂ R2

of the general form

G℘ = ∂2x +

A′(x)

A(x)∂x +

1

A(x)2℘z,

(x ∈ R

+, z ∈ (a, b))

where ℘z = 1r(z)

(p(z)∂2z + p′(z)∂z

)belongs to a class of Sturm-Liouville differential operators on the

interval (a, b), −∞ ≤ a < b ≤ ∞. It will be assumed that the coefficients p, r are such that p, r > 0,

p, r are locally absolutely continuous on the interior of I and the endpoint a is regular or entrance (cf.

Equation (A.1) of the Appendix). As in the previous sections, the coefficient A(x) is assumed to satisfy

the conditions (2.2) and limx→∞A(x) = ∞.

In what follows we set I = [a, b) if b is an exit or natural endpoint of ℘ and I = [a, b] if the endpoint

b is regular or entrance. We shall write M = R+0 × I and ω℘

(d(x, z)

)= A(x)dx r(z)dz.

Let ψη be the unique solution of the boundary value problem −℘z(u) = ηu, u(a) = 1, (pu′)(a) = 0

(cf. Lemma A.1). The eigenfunction expansion of the operator ℘ with Neumann boundary conditions (cf.

Proposition A.2) yields the integral transform

(J℘g)(η) :=∫ b

a

g(z)ψη(z) r(z)dz, (J −1℘ ϕ)(z) :=

∫

R+0

ϕ(η)ψη(z)σ(dη)

which is an isometric isomorphism between L2(r) ≡ L2((a, b), r(z)dz) and L2(R+0 ,σ).

If the endpoint b is regular, entrance or exit (cf. Proposition A.2), then the spectral measure σ is

discrete and the inverse integral transform is written as (J −1℘ ϕ)(z) =

∑∞k=1

1‖ψηk

‖2 ϕ(ηk)ψηk(z), where

the ηk are eigenvalues of ℘. In these conditions, the application of the eigenfunction expansion to functions

h(x, z) ∈ L2(M,ω℘) yields the decomposition

L2(M,ω℘) =

∞⊕

k=1

H℘ηk, H℘

η := {ψη(z)v(x) | v ∈ L2(A)}.

A similar expansion also holds if b is natural, with the direct sums replaced by direct integrals [18, §7.4].

Note also that if u ∈ H℘η ∩ C∞

c (M) then G℘u = Gηku, where Gη := ∂2x +

A′(x)A(x) ∂x −

ηA(x)2 .

The following result is a counterpart of Proposition 2.3:

Proposition 6.1. For each (λ, η) ∈ C × R+0 , there exists a unique solution w℘λ,η ∈ {ψη(z)v(x) | v ∈

C(R+0 )} of the boundary value problem

−G℘u = λu, u(0, z) = ψη(z), u[1](0, z) = 0.

There exists a locally finite positive Borel measure ρ℘ on (R+0 )

2 such that the map h 7→ F℘h, where

(F℘h)(λ, η) :=

∫

M

h(x, z)w℘λ,η(x, z)ω℘(d(x, z))

(λ, η ≥ 0

), (6.1)

is an isometric isomorphism F℘ : L2(M,ω℘) −→ L2((R+0 )

2,ρ℘) whose inverse is given by

(F−1℘ Φ)(x, z) =

∫

(R+0 )2

Φ(λ, η)w℘λ,η(x, z)ρ℘(d(λ, η)). (6.2)

The convergence of the integral in (6.1) is understood with respect to the norm of L2((R+0 )

2,ρ℘) and the

convergence of the integral in (6.2) is understood with respect to the norm of L2(M,ω℘).

22

If b is regular, entrance or exit, then ρ℘(Λ1 × Λ2) =∑ηk∈Λ2

1‖ψηk

‖2ρ℘ηk(Λ1), where ρ℘η is the spectral

measure of (the Neumann self-adjoint extension of) the Sturm-Liouville operator Gη, and the eigenfunc-

tion expansion (6.1)–(6.2) reduces to

F℘ : L2(M,ω℘) −→∞⊕

k=1

L2(R+0 ,ρ

℘ηk), F℘h ≡

((F℘h)(·, η1), (F℘h)(·, η2), . . .

)

(F−1℘ {ϕk})(x, z) =

∞∑

k=1

1

‖ψηk‖2∫

R+0

ϕk(λ)w℘λ,ηk

(x, z)ρ℘ηk(dλ).

Proof. The result for b regular, entrance or exit can be proved in a direct way using the same method as

in Proposition 2.3.

If b is natural, start by considering the operator G℘ on the restricted domain MN = [0, N ] × [a,N ],

where max{0, a} < N < ∞. Applying first the eigenfunction expansion of the Sturm-Liouville operator

℘ on the interval [a,N ] (with boundary condition u′(N) = 0) and then the eigenfunction expansion of

the Sturm-Liouville operators Gη on [0, N ] (also with u′(N) = 0), we obtain a discrete eigenfunction

expansion of the form

(F℘,N h)(λk,N , ηk,N) =

∫

MN

h(x, z)w℘λk,N ,ηk,N(x, z)ω℘(d(x, z))

(F−1℘,N{ck})(x, z) =

∞∑

k=1

ck‖wλk,N ,ηk,N

‖2 wλk,N ,ηk,N(x, z).

Using the techniques of [8], one can show that in the limit N → ∞ this discrete expansion gives rise to an

eigenfunction expansion of the general form (6.1)–(6.2), where ρ℘ is the limiting spectral measure.

As before, we will use the shorthand notation ξ = (x, z), ξ1 = (x1, z1), etc. for points of M .

Proposition 6.2 (Product formula for w℘λ,η). Suppose that there exists a family of measures {π℘z1,z2}z1,z2∈I ⊂P(I) such that

ψη(z1)ψη(z2) =

∫

I

ψη dπ℘z1,z2 (z1, z2 ∈ I, η ∈ supp(σ)). (6.3)

Then for each η ∈ supp(σ) and ξ1, ξ2 ∈ M there exists a positive measure γ℘η,ξ1,ξ2on M such that the

product w℘λ,η(ξ1)w℘λ,η(ξ2) admits the integral representation

w℘λ,η(ξ1)w℘λ,η(ξ2) =

∫

M

w℘λ,η(ξ3)γ℘η,ξ1,ξ2

(dξ3), ξ1, ξ2 ∈M, λ ∈ C, η ∈ supp(σ). (6.4)

Proof. It is straightforward that w℘λ,η(x, z) = ψη(z) ζη(x) vλ,η(x), where ζη(x) := cosh(√η∫ x0

dyA(y)

)and

vλ,η is the solution of

− 1

Bη(Bη v

′)′ = λv, v(0) = 1, (Bηv′)(0) = 0

where Bη(x) = A(x)ζη(x)2. Arguing as in the proof of Proposition 3.1, we deduce that the product

formula (6.4) holds for the positive measures

γ℘η,ξ1,ξ2

(dξ3) =ζη(x1)ζη(x2)

ζη(x3)ν℘η,ξ1,ξ2

(dξ3)

where ν℘η,ξ1,ξ2:= π

[η]x1,x2 ⊗ π℘z1,z2 and π

[η]x1,x2 is the measure of the product formula for vλ,η.

Sufficient conditions for the existence of {π℘z1,z2}z1,z2∈I ⊂ P(I) such that (6.3) holds have been deter-

mined by various authors [4, pp. 311–314], [5, pp. 234–245], [44]. We highlight the following examples:

23

Example 6.3. (a) If I = [0, π/2] and the coefficients p = r satisfy

p(π2− z

)= p(z),

p′

pdecreasing on

[0,π

4

],

p′

p(z) = 2α0 cot(2z) + α1(z) (6.5)

where α0 > 0 and α1 ∈ C∞[0, π/2] satisfies α1(0) = 0, then there exists {π℘z1,z2}z1,z2∈I ⊂ P(I) satisfying

(6.3).

(b) With I, α0 and α1 as above, the same is true if we replace (6.5) by

p(2j+1)(π2

)= 0 for j = 0, 1, 2, . . . ,

p′

pdecreasing on

[0,π

2

],

p′

p(z) = α0 cot(z) + α1(z).

(c) For I = [a,∞), let

α(z) =

∫ z

c

√r(ζ)

p(ζ)dζ, α−1 its inverse function , R(y) =

√p(α−1(y)) r(α−1(y))

where a < c <∞ is a fixed point. If α(∞) = ∞ and the function R′

R is decreasing and nonnegative, then

there exists {π℘z1,z2}z1,z2∈I ⊂ P(I) satisfying (6.3).

Definition 6.4. Suppose that there exists {π℘z1,z2}z1,z2∈I ⊂ P(I) such that (6.3) holds, and let η ∈supp(σ), λ ≥ 0 and µ, ν ∈ MC(M). The measure

(µ ∗η,℘

ν)(·) =

∫

M

∫

M

ν℘η,ξ1,ξ2

(·)µ(dξ1) ν(dξ2)

is called the Gη-convolution of the measures µ and ν. The functions

(F℘ µ)(λ, η) =

∫

M

w℘λ,η(ξ)

ζη(x)µ(dξ) and (T µ

η,℘h)(ξ) :=

∫

M

h d(δξ ∗η,℘

µ)

are, respectively, the G℘-Fourier transform of the measure µ and the Gη-translation of a function h by µ.

Unsurprisingly, the Gη-convolution shares many properties with the ∆k-convolution studied in the

previous sections, among which the following:

Proposition 6.5. Assume that there exists {π℘z1,z2}z1,z2∈I ⊂ P(I) such that (6.3) holds. Assume also

that e−t···∈ L2(R+0 ,σ) for all t > 0.

(a) For each η ∈ supp(σ), the space (MC(M), ∗η,℘

), equipped with the total variation norm, is a com-

mutative Banach algebra over C whose identity element is the Dirac measure δ(0,a). Moreover, the

subset P(M) is closed under the Gη-convolution.

(b)(F℘(µ ∗

η,℘ν))(λ, η) = (F℘ µ)(λ, η) ·(F℘ ν)(λ, η) for all λ ≥ 0 and η ∈ supp(σ).

(c) Each measure µ ∈ MC(M) is uniquely determined by F℘ µ.

Set Σ := supp(σ) if I = [a, b] and set Σ := R+0 if I = [a, b). In the latter case, assume also that

limz↑b ψη(z) = 0 for all η > 0. Then:

(d) Let {µn} be a sequence of measures belonging to M+(M) whose G℘-Fourier transforms are such

that

(F℘ µn)(λ, η) −−−−→n→∞

f(λ, η) pointwise in (λ, η) ∈ R+0 × Σ

for some real-valued function f such that

{f(·, 0) is continuous at a neighbourhood of zero if b is regular or entrance

f is continuous at a neighbourhood of (0, 0) if b is exit or natural.

Then µnw−→ µ for some measure µ ∈ M+(M) such that F℘ µ ≡ f .

24

(e) For each η ∈ Σ the mapping (µ, ν) 7→ µ ∗η,℘

ν is continuous in the weak topology.

(f) If h ∈ Cb(M) (respectively C0(M)) then Tµη,℘h ∈ Cb(M) (resp. C0(M)) for all µ ∈ MC(M).

(g) Let 1 ≤ p ≤ ∞, µ ∈ M+(M) and h ∈ Lpη,℘ := Lp(M,Bη(x)dx r(z)dz

). The Gη-translation

(T µη,℘h)(x) is a Borel measurable function of x ∈M , and we have

‖T µη,℘h‖Lp

η,℘≤ ‖µ‖·‖h‖Lp

η,℘.

Proof. We will only prove (c), (d) and (g) because the proof of parts (a)–(b) and (e)–(f) are analogous

to those of the corresponding results for the ∆k-convolution and translation.

(c) Let µ be such that (F℘ µ)(λ, η) = 0 for λ ≥ 0 and η ∈ supp(σ). For f ∈ Cc(R+0 ) and g ∈ Cc(I)

we have∫

M

f(x) g(z)µ(d(x, z)) = limt↓0

∫

M

f(x)

∫

R+0

e−tη(J℘g)(η)ψη(z)σ(dη)µ(d(x, z))

= limt↓0

∫

R+0

e−tη (J℘g)(η)∫

M

f(x)µη(dx)σ(dη)

= 0

where the measure µη is defined as∫R

+0f dµη =

∫M f(x)ψη(z)µ

(d(x, z)

), and the argument in the proof

of Proposition 3.6(ii) yields the last equality. The conclusion that µ = 0 also follows in the same way.

(d) For I = [a, b], the proof is identical to that of Proposition 3.6(iv). For the case I = [a, b), fix

ε > 0 and notice that the hypothesis (together with the integral mean value theorem) ensures that we

can pick δ > 0 such that ∣∣∣∣2

δ

∫

Vδ

(f(0, 0)− f(λ, η)

)d(λ, η)

∣∣∣∣ < ε

where Vδ := {(λ, η) ∈ (R+0 )

2 | λ2 + η2 < δ}. In addition, we know that limz→∞ ψη(z) = 0 and

limx→∞ vλ,η(x) = 0 for all λ, η > 0 (Lemma A.4(b)), and thus there exist 0 < β1 < ∞ and a < β2 < b

such that ∫

Vδ

(1− ψη(z)vλ,η(x)

)d(λ, η) ≥ δ

2for all (x, z) ∈M \ [0, β1]× [a, β2].

Hence

µn(M \ [0, β1]× [a, β2]

)≤ 2

δ

∫

M\[0,β1]×[a,β2]

∫

Vδ

(1− ψη(z)vλ,η(x)

)d(λ, η)µn(dξ)

≤ 2

δ

∫

Vδ

((F℘ µn)(0, 0)− (F℘ µn)(λ, η)

)d(λ, η)

and consequently

lim supn→∞

µn(M \ [0, β1]× [a, β2]

)≤ 2

δlim supn→∞

∫

Vδ

((F℘ µn)(0, 0)− (F℘µn)(λ, η)

)d(λ, η)

=2

δ

∫

Vδ

(f(0, 0)− f(λ, η)

)d(λ, η)

< ε.

This shows that {µn} is a tight sequence of measures. Applying Prokhorov’s theorem and the reasoning

from the proof of Proposition 3.6(iv), the desired conclusion follows.

(g) It suffices to prove that the generalized translation operator (T z0℘ g)(z) :=

∫I g dπ

℘z,z0 is a con-

traction operator on Lp(I, r) (we then obtain the result by arguing as in Proposition 3.8(c)). Let ⋄℘

be

the convolution determined by the product formula (6.3) and let {κt}t≥0 the ⋄℘

-Gaussian convolution

semigroup generated by ℘. For g ∈ Cc(I) we have∫

I

g(z)(µt ⋄

℘δz1 ⋄℘ δz2

)(dz) = lim

s↓0

∫

I

(T κs℘ g)(z)

(µt ⋄

℘δz1 ⋄℘ δz2

)(dz)

25

= lims↓0

∫

R+0

(J℘g)(η) e−(t+s)η ψη(z1)ψη(z2)σ(dη)

=

∫

I

g(z) qt(z1, z2, z) r(z)dz

where qt(z1, z2, z3) :=∫R

+0e−tη ψη(z1)ψη(z2)ψη(z3)σ(dη). Consequently, for f, g ∈ Cc(I) we have

⟨T µt ⋄

℘δz0

℘ f, g⟩L2(I,r)

=

∫ b

a

∫ b

a

g(z1)qt(z0, z, z1)r(z1)dz1 f(z)r(z)dz =⟨f, T µt ⋄

℘δz0

℘ g⟩L2(I,r)

and by continuity it follows that⟨T z0℘ f, g

⟩L2(I,r)

=⟨f, T z0

℘ g⟩L2(I,r)

. Since T z0℘ is clearly a contraction

on L∞(I, r), by duality we find that it is also a contraction on L1(I, r); hence, by interpolation, it is a

contraction on Lp(I, r) for 1 ≤ p ≤ ∞.

Notions such as infinite divisibility and convolution semigroups with respect to the Gη-convolution

can also be defined like in the previous section, giving rise to a Lévy-Khintchine type representation and

to Feller semigroups on C0(M). The details are left to the reader.

Remark 6.6. The above result on the existence of a product formula for the functions w℘λ,η can be

interpreted in the context of the theory of multiparameter Sturm-Liouville spectral problems.

First we recall some known results. Consider the system of Sturm-Liouville equations

− (pmu′m)

′(xm) + (qmum)(xm) =

N∑

n=1

λn(rmnum)(xm) (m = 1, . . . , N) (6.6)

where N ∈ N and am ≤ xm ≤ bm, together with boundary conditions at the endpoints am and bm of the

form

um(am) cos(ϑm) = u′m(am) sin(ϑm), um(bm) cos(ϑ′m) = u′m(bm) sin(ϑ′m) (m = 1, . . . , N). (6.7)

Let us assume that the intervals Im = [am, bm] are bounded, the functions pm, qm, rmn are sufficiently

well-behaved and r(x) = det{rmn(xm)} > 0 for x = (x1, . . . , xN ) ∈ ∏Nm=1 Im. If λ = (λ1, . . . , λN ) is

chosen such that for each m there exists a nontrivial solution um(xm;λ) of (6.6)–(6.7), then the function

u(x;λ) =∏Ni=1 um(xm;λ) is said to be an eigenfunction of the system (6.6)–(6.7) corresponding to the

eigenvalue λ.

By the completeness theorem for multi-parameter eigenvalue problems [17], the following Fourier-like

expansion holds:

h(x) =∑

k

(Fh)(λ(k))u(x;λ(k)), (6.8)

where

(Fh)(λ(k)) :=

∫ b

a

h(x)u(x;λ(k)) r(x)dx,

λ(k) are the eigenvalues of (6.6)–(6.7), and∫ ba=

∫ b1a1. . .

∫ bNaN

.

Similar results have been established for (singular) systems where some of the intervals [am, bm] are

unbounded; in this case, the sum in (6.8) is, in general, replaced by a Stieltjes integral with respect to

a spectral function [8, 9] However, compared to one-dimensional Sturm-Liouville operators, much less is

known regarding the spectral properties of such singular systems [1, 41].

Returning to our eigenfunction expansion (6.1)–(6.2) for the elliptic operator G℘, we can now rein-

terpret it as a Fourier-like expansion for the system of differential equations (6.6) with N = 2, x1 ∈ R+0 ,

x2 ∈ [a, b], λ1 = λ, λ2 = η, p1 = r11 = A, p2 = p, r22 = r, r12 = 1A and q1 = q2 = r21 = 0.

In the theory of product formulas and convolutions associated with one-dimensional Sturm-Liouville

equations −(pu′)′+qu = λru, a crucial requirement is that the measures of the product formula should not

depend on the spectral parameter λ, cf. e.g. [10, 27, 43]. Similarly, the measures of product formula (6.4)

26

for the generalized eigenfunctions w℘λ,η do not depend on one of the spectral parameters (the measures

γη,ξ1,ξ2are independent of λ); this is a fundamental property which (as we saw above) enables us to

develop the theory of Gη-convolutions. This suggests that the natural way to introduce the notion of a

product formula for a general Sturm-Liouville system (6.6) (regular or singular, with suitable boundary

conditions) is as follows:

Let 1 ≤ s ≤ N . The system (6.6) is said to admit a (λ1, . . . , λs)-product formula if for each x(1), x(2) ∈I :=

∏Nm=1 Im there exists a positive measure γ

λs+1,...,λN

x(1),x(2) on I such that the product u(x(1);λ)u(x(2);λ)

admits the representation

u(x(1);λ)u(x(2);λ) =

∫

I

u(x;λ)γλs+1,...,λN

x(1),x(2) (dx), λ1, . . . , λN ≥ 0. (6.9)

As far as we are aware, this paper is the first to establish the existence of product formulas of the

form (6.9) for a general family of nontrivial Sturm-Liouville systems of the form (6.6); here, the word

‘nontrivial’ means that rmn 6= 0 for some m 6= n. (The few previous results on product formulas of the

form (6.9) only apply to very special cases where the measure is independent of all spectral parameters,

see [33, 46].) Developing a general theory of product formulas for nontrivial systems of Sturm-Liouville

equations is an interesting problem which is left open for further investigation.

Appendix: Generalized convolution structures for Sturm-Liouville

operators

One-dimensional convolutions associated with Sturm-Liouville operators have been extensively studied in

recent work of the authors [43, 44]. For convenience, in this appendix we summarize some fundamental

results on this topic.

Let

ℓ(u)(x) :=1

r(x)

(−(pu′)′(x) + q(x)u(x)

), x ∈ (a, b) ⊂ R

be a Sturm-Liouville expression whose coefficients are such that p, r > 0 on (a, b), p, r are locally absolutely

continuous and q is locally integrable on (a, b).

Consider the integrals

Ia =

∫ c

a

∫ y

a

dx

p(x)

(r(y) + q(y)

)dy, Ja =

∫ c

a

∫ c

y

dx

p(x)

(r(y) + q(y)

)dy

Ib =

∫ b

c

∫ b

y

dx

p(x)

(r(y) + q(y)

)dy, Jb =

∫ b

c

∫ y

c

dx

p(x)

(r(y) + q(y)

)dy.

The Feller boundary classification for ℓ is as follows: the endpoint e ∈ {a, b} is said to be:

regular if Ie <∞, Je <∞;

exit if Ie <∞, Je = ∞;

entrance if Ie = ∞, Je <∞;

natural if Ie = ∞, Je = ∞ (A.1)

(the classification is independent of the choice of c). Throughout the appendix we assume that the

endpoint a is regular or entrance.

Lemma A.1. The boundary value problem

ℓ(v) = λv (a < x < b, λ ∈ C), v(a) = 1, (pv′)(a) = 0

has a unique solution vλ(·). Moreover, λ 7→ vλ(x) is, for fixed x, an entire function of exponential type.

Proof. See [44, Lemma 2.1]. (The result of [44] is stated for the case q ≡ 0, but the general case can be

proved in a similar way.)

27

Proposition A.2. The operator L : D(L) −→ L2(r) ≡ L2((a, b), r(z)dz

), where

D(L) =

{u ∈ L2(r)

∣∣ u, u′∈ ACloc(a, b), ℓ(u) ∈ L2(r), (pu′)(a) = 0}

if b is exit or natural{u ∈ L2(r)

∣∣ u, u′∈ ACloc(a, b), ℓ(u) ∈ L2(r), (pu′)(a) = (pu′)(b) = 0}

otherwise

Lu = ℓ(u), u ∈ D(L)

is a positive self-adjoint operator. There exists a locally finite positive Borel measure ρℓ on R+0 such that

the integral operator Fℓ : L2(r) −→ L2(R+0 ,ρℓ) defined by

(Fℓ f)(λ) :=∫ b

a

f(x) vλ(x) r(x)dz (A.2)

is an isometric isomorphism whose inverse is given by

(F−1ℓ ϕ)(x) =

∫

R+0

ϕ(λ) vλ(x)ρℓ(dλ). (A.3)

(The convergence of the integrals above is understood with respect to the norm of L2(R+0 ,ρℓ) and L2(r)

respectively.) The operator Fℓ is a spectral representation of L, i.e. we have

D(L) ={f ∈ L2(r)

∣∣∣∣∫

R+0

λ2∣∣(Fℓf)(λ)

∣∣2ρℓ(dλ) <∞}

(Fℓ(Lf)

)(λ) = λ ·(Fℓf)(λ), f ∈ D(L).

Moreover, if f ∈ D(L) then f(x) =∫R

+0(Fℓ f)(λ) vλ(x)ρℓ(dλ) for all x ∈ (a, b), where the integral

converges absolutely and locally uniformly.

If b is regular, entrance or exit, then L has purely discrete spectrum 0 = λ1 < λ2 ≤ . . .→ ∞, and the

isomorphism (A.2)–(A.3) reduces to

Fℓ : L2(r) −→ ℓ2(

1‖vλk

‖2

), Fℓ f ≡

((Fℓ f)(λ1), (Fℓ f)(λ2), . . .

), (F−1

ℓ {ck})(x) =∞∑

k=1

ck vλk(x)

‖vλk‖2

where ℓ2(

1‖vλk

‖2

)denotes the weighted sequence space whose norm is ‖{ck}‖ =

(∑∞k=1

|ck|2‖vλk

‖2

)1/2.

Proof. See [44, Proposition 2.5 and Lemma 2.6], [30, Section 5].

Proposition A.3. The self-adjoint operator −L is the generator of a Markovian semigroup {e−tL}t≥0

on L2(r). For t > 0, the operators e−tL are given by

(e−tLh)(x) =

∫ b

a

h(y) pℓ(t, x, ξ) r(ξ)dξ(h ∈ L2(r), x ∈ (a, b)

)(A.4)

where the kernel is defined by the integral

pℓ(t, x1, x2) =

∫

R+0

e−tλ vλ(x1) vλ(x2)ρℓ(dλ)(t > 0, x1, x2 ∈ (a, b)

). (A.5)

The right hand side of (A.5) is (for fixed t > 0) absolutely and uniformly convergent on compact squares

of (a, b)× (a, b).

Suppose also that a is regular and b is exit or natural. Then the integral in (A.5) converges absolutely

and uniformly on compact squares of [a, b)× [a, b). Moreover, the restriction of e−tL to L2(r) ∩ C0[a, b)

extends into a strongly continuous contraction semigroup on C0[a, b) which can be represented by the right

hand side of (A.4), which is convergent for all h ∈ C0[a, b) and x ∈ [a, b).

Proof. See [44, Proposition 2.7], [20].

28

Next we restrict our attention to the case q ≡ 0 and state some further properties of the generalized

eigenfunctions vλ(x).

Lemma A.4. (a) If q ≡ 0 and x 7→ p(x)r(x) is an increasing function, then |vλ(x)| ≤ 1 for all a ≤ x < b

and λ ≥ 0.

(b) Let S(ξ) :=√p(γ−1(ξ)) r(γ−1(ξ)), where γ(x) =

∫ xc

√r(y)p(y)dy and γ−1 is its inverse function. (Here

c ∈ (a, b) is a fixed point; if√

r(y)p(y) is integrable near a, then we may also take c = a.) Assume that

q ≡ 0, γ(b) =∫ bc

√r(y)p(y)dy = ∞, and there exists η ∈ C1(γ(a),∞) such that η ≥ 0, the

functions φη := S′

S − η, ψη := 12η

′ − 14η

2 + S′

2S ·η are both decreasing on (γ(a),∞) and

φη satisfies limξ→∞ φη(ξ) = 0.

(A.6)

Then the following assertions are equivalent:

• limx↑b p(x)r(x) = ∞;

• limx↑b vλ(x) = 0 for all λ > 0.

Proof. See [44, Lemma 2.3 and Proposition 3.6].

Theorem A.5 (Product formula for vλ). Assume that (A.6) holds. Then there exists a family of measures

{πx1,x2}x1,x2∈[a,b) ⊂ P [a, b) such that we have

vλ(x1) vλ(x2) =

∫

[a,b)

vλ(x3)πx1,x2(dx3) for all x1, x2 ∈ [a, b), λ ∈ C.

If p ≡ r and a = γ(a) = 0, then supp(πx1,x2) ⊂ [|x1 − x2|, x1 + x2].

Proof. See [44, Section 4 and Subsection 5.3].

Proposition A.6. Assume that (A.6) holds. Let Fℓ be the ℓ-Fourier transform of measures defined by

(Fℓ µ)(λ) :=∫

[a,b)

vλ(x)µ(dx) (µ ∈ P [a, b), λ ≥ 0). (A.7)

Then:

(i) Fℓ µ is continuous on R+0 . Moreover, if the family {µj} ⊂ MC[a, b) is tight and uniformly bounded,

then {Fℓµj} is equicontinuous on R+0 .

(ii) Let µ1, µ2 ∈ MC[a, b). If Fℓ µ1 ≡ Fℓ µ2, then µ1 = µ2.

(iii) Let {µn} ⊂ M+[a, b), µ ∈ M+[a, b), and suppose that µnw−→ µ. Then

Fℓ µn −−−−→n→∞

Fℓ µ uniformly on compact sets.

Proof. See [44, Proposition 5.2].

The ℓ-convolution and the ℓ-translation operator are respectively defined by

(µ ⋄ℓν)(dξ) :=

∫

[a,b)

∫

[a,b)

πx,y(dξ)µ(dx) ν(dy), µ, ν ∈ MC[a, b)

(T µℓ f)(x) :=

∫

[a,b)

f d(δx ⋄ℓµ), µ ∈ MC[a, b), x ∈ [a, b), f ∈ Lp(r)

where Lp(r) ≡ Lp((a, b), r(z)dz

)(1 ≤ p ≤ ∞).

29

Proposition A.7. Assume that (A.6) holds.

(a) µ = µ1 ⋄ℓµ2 if and only if (Fℓ µ)(λ) = (Fℓ µ1)(λ) ·(Fℓ µ2)(λ) for all λ ≥ 0 (µ, µ1, µ2 ∈ MC[a, b)).

(b) The ⋄ℓ

convolution is weakly continuous: if µnw−→ µ and νn

w−→ ν, then µn ⋄ℓνn

w−→ µ ⋄ℓν.

(c) If f ∈ C2c [a, b) with f ′ ∈ Cc(a, b), then

∫

[a,b)

f d(δx ⋄ℓµ) =

∫

R+0

(Fℓ f)(λ) (Fℓ µ)(λ) vλ(x)ρℓ(dλ) for all µ ∈ MC[a, b), x ∈ (a, b).

(d) Suppose that limx↑b p(x)r(x) = ∞, and let µ ∈ MC[a, b). Then δx ⋄ℓµ

v−→ 0 as x ↑ b, where 0 is the

zero measure.

(e) Let 1 ≤ p ≤ ∞ and µ ∈ M+[a, b). The ℓ-translation T µℓ is a bounded operator on Lp(r) such that

‖T µℓ f‖Lp(r) ≤ ‖µ‖·‖f‖Lp(r) for all f ∈ Lp(r).

(f) Let 1 ≤ p1, p2 ≤ ∞ with 1p1

+ 1p2

≥ 1, and let f ∈ Lp1(r), g ∈ Lp2(r). Then the ℓ-convolution

(f ⋄ℓg)(x) :=

∫ b

a

(T yℓ f)(x) g(y) r(y)dy

(where T yℓ ≡ T δy

ℓ ) is well-defined and satisfies

‖f ⋄ℓg‖Ls(r) ≤ ‖f‖Lp1(r)‖g‖Lp2(r), where s =

1

1/p1 + 1/p2 − 1.

Proof. See [43, Corollary 5.2 and Proposition 6.4], [44, Proposition 5.4 and Lemma 5.5].

Proposition A.8. If (A.6) holds with p ≡ r and a = γ(a) = 0, then(R

+0 , ⋄ℓ

)is a commutative hyper-

group (in the sense of [5, 24]) with identity element δ0 and trivial involution, i.e. the following axioms

hold:

•(R

+0 , ⋄ℓ

), equipped with the total variation norm, is a commutative Banach algebra over C whose

identity element is the measure δ0;

• If µ, ν ∈ P(R+0 ), then µ ⋄

ℓν ∈ P(R+

0 );

• (µ, ν) 7→ µ ⋄ℓν is continuous in the weak topology of measures;

• (x1, x2) 7→ supp(δx1 ⋄ℓ δx2) is continuous from R+0 ×R

+0 into the space of compact subsets of R+

0 , and

we have 0 ∈ supp(δx1 ⋄ℓ δx2) if and only if x1 = x2.

Proof. See [44, Subsection 5.3].

The measures {µt}t≥0 ⊂ P [a, b) are said to be an ℓ-convolution semigroup if

µs ⋄ℓµt = µs+t for all s, t ≥ 0, µ0 = δa and µt

w−→ δa as t ↓ 0.

Proposition A.9. Let ℓ be a Sturm-Liouville expression with p ≡ r, q ≡ 0 and a = γ(a) = 0. Suppose

that (A.6) holds and that limx↑b p(x) = ∞.

(a) For t > 0, let αℓt be the measure defined by αℓt(dx) := pℓ(t, 0, x)r(x)dx, where pℓ(t, x1, x2) is the kernel

(A.5). Set αℓ0 = δ0. Then {αℓt}t≥0 is an ℓ-convolution semigroup such that

limt↓0

1

tαℓt [ε,∞) = 0 for every ε > 0.

(b) Let ψ(λ) be a function which can be written as

ψ(λ) = cλ+

∫

R+

(1− vλ(x)) τ(dx) (λ ≥ 0)

30

for some c ≥ 0 and some positive measure τ on R+ such that τ is finite on the complement of any

neighbourhood of 0 and satisfies∫R+(1− vλ(x)) τ(dx) <∞ for λ ≥ 0. Then there exists an ℓ-convolution

semigroup {µt}t≥0 such that (Fℓ µt)(λ) = e−tψ(λ). Moreover, there exists a constant C > 0 independent

of λ such that

ψ(λ) ≤ C(1 + λ) for all λ ≥ 0. (A.8)

Proof. Part (a) and the first statement in part (b) follow from [43, Theorem 7.3 and Propositions 7.11–

7.12].

To prove the estimate (A.8), start by picking λ1 > 0. We know that limx↑b vλ1 (x) = 0 (Lemma

A.4(b)), hence there exists β ∈ (a, b) such that |vλ1(x)| ≤ 12 for all β ≤ x < b. Combining this with

Lemma A.4(a), we deduce that for all λ ≥ 0 we have

n

∫

[β,b)

(1− vλ(x)

)µ1/n(dx) ≤ 2n

∫

[β,b)

µ1/n(dx)

≤ 4n

∫

[β,b)

(1− vλ1(x)

)µ1/n(dx)

≤ 4n(1− (Fℓµ1/n)(λ1)

)≤ 4ψ(λ1).

(A.9)

Next, choose λ2 > 0 such that 1 − vλ(x) <12 for all 0 ≤ λ ≤ λ2 and all a < x ≤ β. (This is possible

because of the boundedness of the family of derivatives {∂λv(·)(x)}x∈(a,β], cf. [44, pp. 5–6].) Defining

η1(x) :=∫ xa

1p(y)

∫ ya r(ξ)dξ dy, we obtain

1− vλ2(x) = λ2

∫ x

a

1

p(y)

∫ y

a

vλ2(ξ)r(ξ)dξ dy ≥ λ22η1(x) for all a ≤ x ≤ β.

On the other hand, by Lemma A.4(a) we have 1− vλ(x) ≤ λ∫ xa

1p(y)

∫ ya |vλ(ξ)|r(ξ)dξ dy ≤ λη1(x) for all

x ∈ [a, b) and λ ≥ 0. Consequently,

n

∫

[a,β)

(1− vλ(x)

)µ1/n(dx) ≤ λn

∫

[a,β)

η1(x)µ1/n(dx)

≤ 2λn

λ2

∫

[a,β)

(1− vλ2(x)

)µ1/n(dx)

≤ 2λn

λ2

(1− (Fℓµ1/n)(λ2)

)≤ 2λ

λ2ψ(λ2).

(A.10)

Combining (A.9) and (A.10) one sees that for all n ∈ N and λ ≥ 0 we have n(1− e−ψ(λ)/n) ≤ C(1 + λ),

where C = max{4ψµ(λ1),

2λ2ψµ(λ2)

}. The conclusion follows by taking the limit as n→ ∞.

Acknowledgements

The first and third authors were partially supported by CMUP, which is financed by national funds

through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the project with reference

UIDB/00144/2020. The first author was also supported by the grant PD/BD/135281/2017, under the

FCT PhD Programme UC|UP MATH PhD Program. The second author was partially supported by the

project CEMAPRE/REM – UIDB/05069/2020 – financed by FCT through national funds.

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